Risk-Sensitive Capital Requirements and Pro-Cyclicality in ... · Risk-Sensitive Capital Requirements and Pro-Cyclicality in Lending Inauguraldissertation zur Erlangung des akademischen

Risk-Sensitive Capital Requirements

and Pro-Cyclicality in Lending

Inauguraldissertation zur Erlangung des akademischen Grades

eines Doktors der Wirtschaftswissenschaften

der Universitat Mannheim

vorgelegt an der Fakultat fur Betriebswirtschaftslehre

der Universitat Mannheim

Volker Sygusch

Mannheim, im Herbstsemester 2010

ii

Dekan: Dr. Jurgen M. Schneider

Referent: Prof. em. Dr. Dr. h.c. Wolfgang Buhler

Korreferent: Prof. Dr. Peter Albrecht

Tag der mundlichen Prufung: 24. Januar 2011

Contents

List of Figures viii

List of Tables xii

List of Abbreviations xiii

List of Frequently Used Symbols xvi

Acknowledgements xix

I Introduction 1

1 The Issue of Pro-Cyclicality 3

1.1 The Pro-Cyclicality of Lending . . . . . . . . . . . . . . . . . . . . . 6

1.2 The Basel I Accord and Pro-Cyclicality . . . . . . . . . . . . . . . . . 8

1.2.1 The Introduction of Basel I and the 1990/91 Credit Crunch . 8

1.2.2 Theoretical Research on the Pro-Cyclicality of Basel I . . . . . 10

1.3 The Basel II Accord and Pro-Cyclicality . . . . . . . . . . . . . . . . 11

1.3.1 The Basel II Accord . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Pro-Cyclicality of Required Minimum Capital . . . . . . . . . 13

1.3.3 Pro-Cyclicality and Capital Buffers . . . . . . . . . . . . . . . 16

1.3.4 Pro-Cyclicality and Loan-Rating Systems . . . . . . . . . . . . 20

1.3.5 Further Theoretical Research on the Pro-Cyclicality of Basel II 22

1.4 Further Regulations and Pro-Cyclicality . . . . . . . . . . . . . . . . 23

iii

iv CONTENTS

1.5 Risk-Taking, Systemic Risk, and Capital Requirements . . . . . . . . 24

1.5.1 Basel I and Risk-Taking . . . . . . . . . . . . . . . . . . . . . 24

1.5.1.1 Empirical Evidence and Criticism . . . . . . . . . . . 25

1.5.1.2 Theoretical Criticism . . . . . . . . . . . . . . . . . . 26

1.5.2 Basel II and Risk-Taking . . . . . . . . . . . . . . . . . . . . . 28

1.5.2.1 The Three Pillars of Basel II . . . . . . . . . . . . . 28

1.5.2.2 Systemic Risk . . . . . . . . . . . . . . . . . . . . . . 29

2 Framework of the Analysis 31

2.1 Exogenous Shocks and the Business Cycle . . . . . . . . . . . . . . . 31

2.1.1 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.2 Expectation Shocks . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.3 Interference of Different Shocks . . . . . . . . . . . . . . . . . 32

2.2 Different Types of Cyclical Behavior . . . . . . . . . . . . . . . . . . 33

2.3 The Model’s Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 The Basic Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.2 Fixed Bank’s Equity Capital and Uninsured Deposits . . . . . 36

2.3.3 The Bank’s Risk-Neutrality . . . . . . . . . . . . . . . . . . . 38

2.3.4 Market Power in Banking . . . . . . . . . . . . . . . . . . . . 38

II Bernoulli Distributed Loan Redemptions 43

3 Theoretical Analysis 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 The Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 The Repayment of Deposits . . . . . . . . . . . . . . . . . . . 49

3.2.3 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 The Bank’s Expected Pay-off . . . . . . . . . . . . . . . . . . 63

CONTENTS v

3.3 Equilibrium and Sensitivities . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 The Equilibrium without Regulation . . . . . . . . . . . . . . 65

3.3.1.1 General Characterization . . . . . . . . . . . . . . . 65

3.3.1.2 Characterization if Case 1 Prevails . . . . . . . . . . 67

3.3.1.3 Characterization if Case 2 with `∗ = 0 Prevails . . . 68


3.3.1.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . 79

3.3.2 The Equilibrium with Regulation by Fixed Risk Weights . . . 80



3.3.2.3 Characterization if Case 2 with `S = 0 Prevails . . . 87


3.3.3 The Equilibrium with Regulation by a Value-at-Risk Approach 95


3.3.3.2 Characterization of Case 1 if lV = α2

α1+α2Prevails . . 101


3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Numerical Analysis of Regulatory Impacts 117

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2.1 Total Loan Volumes and Pro-Cyclical Effects . . . . . . . . . . 119

4.2.2 Tighter Confidence Levels and Total Volumes . . . . . . . . . 123

4.2.3 Single Loans and Pro-Cyclical Effects . . . . . . . . . . . . . . 124

4.2.4 Return Volatilities of Total Loans and Deposits . . . . . . . . 127

4.2.5 Deposit Interest Rates . . . . . . . . . . . . . . . . . . . . . . 131

4.3 Credit Risk Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.1 Total Volumes and Cyclical Dampening . . . . . . . . . . . . . 132

4.3.2 Single Loans and the Role of Constant Risk Weights . . . . . 141

vi CONTENTS

4.3.3 Flexible Risk Weights . . . . . . . . . . . . . . . . . . . . . . . 142

4.3.4 Effects under the Value-at-Risk Approach . . . . . . . . . . . 142

4.4 Productivity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.4.1 Total Volumes and Cyclical Dampening . . . . . . . . . . . . . 146

4.4.2 Effects under the Standardized Approach . . . . . . . . . . . . 148

4.4.3 Effects under the Value-at-Risk Approach . . . . . . . . . . . 151

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

III Normally Distributed Loan Redemptions 161

5 A Basis for the Normal Distribution 163

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.2 Firms and Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.3 The m-Dependence Structure and Its Limit Distribution . . . . . . . 168

5.4 The Limit Distributions of the Aggregate Loans . . . . . . . . . . . . 170

6 The One-Period Model 177

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.2 The Bank’s Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.3 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.4 Equilibrium without Regulation . . . . . . . . . . . . . . . . . . . . . 187

6.5 Equilibrium with Regulation by a Value-at-Risk Approach . . . . . . 191

6.6 Numerical Analysis of Regulatory Impacts . . . . . . . . . . . . . . . 194

6.6.1 Equity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.6.2 Expected Return Shocks . . . . . . . . . . . . . . . . . . . . . 204

6.6.3 Return Volatility Shocks . . . . . . . . . . . . . . . . . . . . . 207

6.6.4 Correlation Shocks . . . . . . . . . . . . . . . . . . . . . . . . 213

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7 The Two-Period Model 221

CONTENTS vii

7.1 Timeline and Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.2 Numerical Analysis of Regulatory Impacts . . . . . . . . . . . . . . . 230

7.2.1 Expected Return Shocks . . . . . . . . . . . . . . . . . . . . . 232

7.2.2 Return Volatility Shocks . . . . . . . . . . . . . . . . . . . . . 237

7.2.3 Return Correlation Shocks . . . . . . . . . . . . . . . . . . . . 239

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

IV Final Remarks 243

Appendix 247

A Proofs to Chapter 3 249

A.1 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

A.1.1 Proof of Result 2 . . . . . . . . . . . . . . . . . . . . . . . . . 249

A.1.2 Derivations concerning Result 3 . . . . . . . . . . . . . . . . . 251

A.1.3 Proof of Result 4 . . . . . . . . . . . . . . . . . . . . . . . . . 253

A.2 Results on Specific Instances of the Equilibrium . . . . . . . . . . . . 254

A.2.1 The Equilibrium without Regulation . . . . . . . . . . . . . . 254

A.2.1.1 Proof of Result 5 . . . . . . . . . . . . . . . . . . . . 254

A.2.1.2 Proof of Result 6 . . . . . . . . . . . . . . . . . . . . 259

A.2.1.3 Proof of Result 7 . . . . . . . . . . . . . . . . . . . . 261

A.2.1.4 Proof of Result 8 . . . . . . . . . . . . . . . . . . . . 264

A.2.1.5 Proof of Result 9 . . . . . . . . . . . . . . . . . . . . 268

A.2.1.6 Proof of Result 10 . . . . . . . . . . . . . . . . . . . 271

A.2.1.7 Proof of Result 11 . . . . . . . . . . . . . . . . . . . 273

A.2.1.8 Proof of Result 12 . . . . . . . . . . . . . . . . . . . 275

A.2.2 The Equilibrium with Regulation by Fixed Risk Weights . . . 276

A.2.2.1 Proof of Result 14 . . . . . . . . . . . . . . . . . . . 276

A.2.2.2 Proof of Result 16 . . . . . . . . . . . . . . . . . . . 279

viii CONTENTS

A.2.2.3 Proof of Result 17 . . . . . . . . . . . . . . . . . . . 282

A.2.2.4 Proof of Result 18 . . . . . . . . . . . . . . . . . . . 283

A.2.2.5 Proof of Result 19 . . . . . . . . . . . . . . . . . . . 284

A.2.2.6 Proof of Result 20 . . . . . . . . . . . . . . . . . . . 287

A.2.2.7 Proof of Result 21 . . . . . . . . . . . . . . . . . . . 288

A.2.3 The Equilibrium with Regulation by a Value-at-Risk Approach 290

A.2.3.1 The Regulatory Constraints . . . . . . . . . . . . . . 290

A.2.3.2 Proof of Result 24 . . . . . . . . . . . . . . . . . . . 291

A.2.3.3 Proof of Result 25 . . . . . . . . . . . . . . . . . . . 293

B Derivations to Chapter 5 297

B.1 Bernoulli Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . 297

B.2 Moments of Aggregate Loan Repayments . . . . . . . . . . . . . . . . 301

C Proofs to Chapter 6 303

C.1 Proof of Result 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

C.2 Proof of Result 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

C.3 Proof of Result 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

D Proofs to Chapter 7 315

D.1 Existence of the Deposit Supply Function in the First Period . . . . . 315

D.2 Proof of Result 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Bibliography 331

Curriculum Vitæ 344

List of Figures

3.1 The Case constraints given a fixed deposit volume . . . . . . . . . . . 52

3.2 The Case constraints given the household’s deposit-supply function . 67

4.1 Total loan volume as a function of the bank’s initial equity: w/o

regulation, StdA with c1 = c2 = 1, VaR at 1.0%, and VaR at 0.5% . . 120

4.2 Total loan volume as a function of the bank’s initial equity: w/o

regulation, StdA with c1 = 0.5 and c2 = 1, and VaR at 0.2% . . . . . 121

4.3 Loan-allocation rate as a function of the bank’s initial equity: w/o

regulation, StdA with c1 = c2 = 1, VaR at 1%, and VaR at 0.5% . . . 122

4.4 Loan-allocation rate as a function of the bank’s initial equity: w/o


4.5 Loan to Firm 1 as a function of the bank’s initial equity: w/o






4.8 Return volatilities as a function of the bank’s initial equity: w/o

regulation, StdA c1 = c2 = 1, and VaR at 1% . . . . . . . . . . . . . . 128

4.9 Return volatilities as a function of the bank’s initial equity: w/o

regulation, StdA c1 = 0.5 and c2 = 1, and VaR at 0.2% . . . . . . . . 129

4.10 Deposit interest rate as a function of the bank’s initial equity: w/o

regulation, StdA with c1 = c2 = 1, StdA with c1 = 0.5 and c2 = 1,

VaR at 1%, VaR at 0.5%, and VaR at 0.2% . . . . . . . . . . . . . . 132

ix

x LIST OF FIGURES

4.11 Total loan volume as a function of shifts in success probabilities: w/o


VaR at 0.5%, and VaR at 0.2% . . . . . . . . . . . . . . . . . . . . . 133

4.12 Total loan volume as a function of shifts in success probabilities: w/o


VaR at 0.5%, and VaR at 1%. . . . . . . . . . . . . . . . . . . . . . . 134

4.13 Deposit interest rate as a function of the correlation . . . . . . . . . . 135

4.14 Total loan volume as a function of the correlation . . . . . . . . . . . 137

4.15 Return volatilities as a function of shifts in success probabilities: w/o

regulation, StdA with c1 = c2 = 1, and with c1 = 0.5 and c2 = 1. . . . 139

4.16 Loan to Firm 1 as a function of the correlation: w/o regulation, StdA

with c1 = c2 = 1, StdA with c1 = 0.5 and c2 = 1, and VaR at 1% . . . 140

4.17 Loan to Firm 1 as a function of the correlation: w/o regulation, StdA

with c1 = c2 = 1, StdA with c1 = 0.5 and c2 = 1, VaR at 0.5%, and

VaR at 0.2% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.18 Return volatilities as a function of shifts in success probabilities . . . 145

4.19 Total loan volume as a function of shifts in productivity: w/o

regulation, StdA with c1 = c2 = 1, StdA with c1 = 0.5, c2 = 1,

VaR at 1%, and VaR at 0.5% . . . . . . . . . . . . . . . . . . . . . . 146

4.20 Total loan volume as a function of shifts in productivity: w/o


VaR at 0.5%, and VaR at 0.2%. . . . . . . . . . . . . . . . . . . . . . 148

4.21 Loan to Firm 1 as a function of its expected gross return: w/o


VaR at 1%, and VaR at 0.5%. . . . . . . . . . . . . . . . . . . . . . . 149

4.22 Loan to Firm 1 as a function of its expected gross return: w/o


VaR at 0.5%, and VaR at 0.2%. . . . . . . . . . . . . . . . . . . . . . 150

4.23 Loan to Firm 2 as a function of shifts in productivity: w/o regulation,

StdA with c1 = c2 = 1, StdA with c1 = 0.5, c2 = 1, VaR at 1%, and

VaR at 0.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

LIST OF FIGURES xi

4.24 Loan to Firm 2 as a function of shifts in productivity: w/o regulation,

StdA with c1 = c2 = 1, StdA with c1 = 0.5, c2 = 1, VaR at 0.5%, and

VaR at 0.2%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.25 Return volatilities as a function of shifts in productivity: w/o

regulation, StdA with c1 = c2 = 1, and VaR at 0.2%. . . . . . . . . . 155

4.26 Return volatilities as a function of shifts in productivity: w/o

regulation, StdA with c1 = c2 = 1, and StdA with c1 = 0.5 and

c2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.1 Correlation structure between projects . . . . . . . . . . . . . . . . . 168

6.1 Illustration of Condition (6.3.4) . . . . . . . . . . . . . . . . . . . . . 183

6.2 Illustration of Condition (6.3.10) . . . . . . . . . . . . . . . . . . . . 186

6.3 Total loan volume as a function of the bank’s initial equity . . . . . . 196

6.4 Deposit interest rate as a function of the bank’s initial equity . . . . . 197

6.5 Portfolio-allocation rate as a function of the bank’s initial equity . . . 198

6.6 Return volatilities as a function of the bank’s initial equity . . . . . . 199

6.7 Probability of full deposit redemption . . . . . . . . . . . . . . . . . . 200

6.8 Loan volume to Sector 1 as a function of the bank’s initial equity . . 201

6.9 Total loan volume as a function of WB, Table 6.2, Panel A . . . . . . 203

6.10 Total loan volume as a function of WB, Table 6.2, Panel B . . . . . . 203

6.11 Total loan volume as a function of µ1 . . . . . . . . . . . . . . . . . . 204

6.12 Total loan volume as a function of µ1, Table 6.2, Panel A . . . . . . . 205

6.13 Return volatilities as a function of µ1 . . . . . . . . . . . . . . . . . . 206

6.14 Total loan volume as a function of σ1 . . . . . . . . . . . . . . . . . . 207

6.15 Total loan volume as a function of σ1, Table 6.2, Panel A . . . . . . . 208

6.16 Loan volume to Sector 2 as a function of σ1 . . . . . . . . . . . . . . 209

6.17 Return volatilities as a function of σ1: w/o regulation, VaR at 1.0%,

and VaR at 0.1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.18 Return volatilities as a function of σ1: w/o regulation, StdA with

c1 = c2 = 1, and StdA with c1 = 0.5, c2 = 1. . . . . . . . . . . . . . . 211

xii LIST OF FIGURES

6.19 Risk effects in equilibrium on deposits . . . . . . . . . . . . . . . . . 212

6.20 Total loan volume as a function of the inter-sectoral correlation . . . 213

6.21 Total loan volume as a function of ρ, Table 6.2, Panel A . . . . . . . 215

6.22 Loan volume to Sector 1 as a function of ρ . . . . . . . . . . . . . . . 216

6.23 Loan volume to Sector 1 as a function of ρ, Table 6.2, Panel A . . . . 217

6.24 Return volatilities as a function of ρ: w/o regulation, VaR at 1.0%,

and VaR at 0.1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.25 Return volatilities as a function of ρ: w/o regulation, StdA with

c1 = c2 = 1, and StdA with c1 = 0.5 and c2 = 1. . . . . . . . . . . . . 219

7.1 Wealth timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.2 Total loan volume L0 (L) as a function of µ1,1 (µ1) . . . . . . . . . . 230

7.3 E0(WB,1) and E(WB) as functions of expectation shocks . . . . . . . . 231

7.4 E0(WB,1) and E0(WB,2) as functions of expectation shocks in t = 0 . . 232

7.5 Expected total loan volume E0(L1) as a function of µ1,1 . . . . . . . . 233

7.6 (Expected) total loan volume as a function of the bank’s (expected)

equity w/o regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.7 (Expected) total loan volume as a function of the bank’s (expected)

equity under VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

7.8 Total loan volume L0 (L) as a function of σ1,1 (σ1) . . . . . . . . . . . 237

7.9 Expected total loan volume E0(L1) as a function of σ1,1 . . . . . . . . 238

7.10 Total loan volume L0 (L) as a function of ρ1 (ρ) . . . . . . . . . . . . 239

7.11 Expected total loan volume E0(L1) as a function of ρ1 . . . . . . . . . 240

A.1 The function f1(RD) and its relation to the identity RD ≡ RD . . . . 251

C.1 Feasible graphs for the function f(RD) . . . . . . . . . . . . . . . . . 312

C.2 The function f(RD) and its relation to the identity RD ≡ RD . . . . . 313

List of Tables

3.1 Parameter values satisfying Conditions (3.3.2) and (3.3.3) . . . . . . . 71

3.2 Optimal choices given Case 4 vs. equilibria . . . . . . . . . . . . . . . 74

3.3 Example of equal projects in the Bernoulli model w/o regulation . . . 80

3.4 Risk weights for corporates in the Standardized Approach . . . . . . 81

3.5 Example illustrating Result 18 . . . . . . . . . . . . . . . . . . . . . . 90

3.6 Constraints by regulation through VaR . . . . . . . . . . . . . . . . . 97

3.7 Example illustrating Result 27 . . . . . . . . . . . . . . . . . . . . . . 110

3.8 Example of equal projects under VaR-based regulation, Bernoulli model113

3.9 Summary of cyclical effects through regulation by fixed risk weights . 115

3.10 Summary of cyclical effects by VaR-based regulation . . . . . . . . . 116

4.1 Parameter values of the base case, Bernoulli model . . . . . . . . . . 118

4.2 Effects on the total loan volume by VaR-based regulation, Bernoulli

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.1 Parameter values of the base case, Bernoulli mixture model . . . . . . 172

5.2 Correlation values of the base case, Bernoulli mixture model . . . . . 173

5.3 Comparison between simulated and limiting distribution . . . . . . . 174

6.1 Parameter values of the base case, normal model . . . . . . . . . . . . 194

6.2 Alternative parameter values, normal model . . . . . . . . . . . . . . 195

xiii

xiv LIST OF TABLES

List of Abbreviations

Art. Article

BCBS Basel Committee of Banking Supervision

cf. compare

Ch. Chapter

EC European Community

EEC European Economic Community

e.g. for example

EMU Economic and Monetary Union of the European Union

Eq. Equation

EU European Union

f and the following page

Fig. Figure

ff and following pages

G-10 Group of Ten (Belgium, Canada, France, Germany, Italy,

Japan, the Netherlands, Sweden, Switzerland, the United

Kingdom and the United States)

G-20 Group of Twenty (Argentina, Australia, Brazil, Canada,

China, France, Germany, India, Indonesia, Italy, Japan,

Mexico, Russia, Saudi Arabia, South Africa, South Korea,

Turkey, the United Kingdom, the United States, and the

European Union)

GDP Gross Domestic Product

IAS International Accounting Standards

ibid. ibidem

i.e. that is

IRB Internal-Ratings-Based

IS Investment/Saving equilibrium

xv

xvi LIST OF ABBREVIATIONS

Iss. Issue

LM Liquidity preference and Money supply

n/a not applicable; not available

No. number

OECD Organization for Economic Co-operation and Development

p. page

PCA Prompt Corrective Action

PD probability of default

PIT point-in-time

pp. pages

Res. Result

S&P Standard & Poor’s

StdA Standardized Approach

TTC through-the-cycle

UK, U.K. United Kingdom of Great Britain and Northern Ireland

US United States

USA United States of America

VaR value-at-risk

Vol. Volume

w/o without

w/r/t with respect to

List of Frequently Used Symbols

bdx,t the bank’s default barrier at t, t = 1, 2, in terms of non-

standardized portfolio returns

bdz ,t the bank’s default barrier at t, t = 1, 2, in terms of standardized

portfolio returns

bsz the bank’s solvency barrier in terms of standardized portfolio

returns (one-period model)

c Cooke ratio

ci risk weight for loan i

Cj feasibility set characterizing the contingent pay-offs of the bank

to its depositor (“Case j”)

Corr(·, ·) correlation operator

D contracted deposit volume (bank debt)

D deposit repayment

e Euler’s number

E(·) expectations operator

i index number for corporate borrowers/sector, i = 1, 2

j index number for the Case Cj

j index number for corporate borrower j belonging to any of

both sectors i

k(·) constraint function under regulation with fixed risk weights ci

L contracted total loan volume

Li contracted volume of the loan granted to borrower i

L total loan repayments

lim limit operator

max maximization operator

n number of borrowers, firms/projects from one sector

pi success probability of firm/project i

pji success probability of firm/project j from sector i

P(·) probability operator

xvii

xviii LIST OF FREQUENTLY USED SYMBOLS

q common success probability of both projects, Bernoulli model

qj the probability of bank solvency/full deposit redemption given

Case j, Bernoulli model

RD contracted gross interest rate on deposits D

Rf gross interest rate on the risk-free asset

Ri contracted gross interest rate on loan/sector-specific loan type

i

si standard deviation of returns from sector i

s1,2 covariance of returns from Sectors 1 and 2

S superscript indicating an equilibrium under regulation by fixed

risk weights ci

t index of time, t = 0, 1, 2 (two-period model)

U(·) the household’s preference function

uH(·) the household’s von Neumann-Morgenstern utility index

v(·, ·) constraint function under regulation by VaR, Bernoulli model

V superscript indicating an equilibrium under regulation by VaR

V(·) variance operator

w(·) constraint function under regulation by VaR, normal model

WB (WB,0) the bank’s initial equity (in the two-period model)

WB the bank’s equity at the end of the period (one-period models)

WB,t the bank’s equity at t, t = 1, 2 (two-period model)

WH (WH,0) the household’s initial wealth (in the two-period model)

WH the household’s wealth at the end of the period (one-period

models)

WH,t the household’s wealth at t, t = 1, 2 (two-period model)

x normally distributed gross return on the bank’s total loan

portfolio

xi normally distributed gross return on the bank’s sector-i loan

portfolio

xt normally distributed gross return on the bank’s total loan

portfolio realized at t, t = 1, 2

Xi Bernoulli random number indicating firm/project i’s success or

failure

Xji Bernoulli random number indicating firm/project j’s success

or failure

zt standardized normally distributed gross return on the bank’s

total loan portfolio realized at t, t = 1, 2

xix

αi gross return on project i in case of success

γ the household’s absolute risk aversion

µj expected marginal gross return on deposits under Case j

µi expected gross return on loans from sector i

ρ correlation between sectors

σj volatility of marginal gross return on deposits under Case j

σi volatility of gross return on loans from sector i

τ regulatory parameter under regulation with value-at-risk

ϕ(·) density function of the standard normal distribution

Φ(·) cumulative distribution function of the standard normal

distribution

` loan-allocation rate/portfolio-allocation rate

N (µ, σ) normal distribution with parameters µ and σ

R the real numbers

∗ superscript indicating laissez-faire equilibrium

xx LIST OF FREQUENTLY USED SYMBOLS

Acknowledgements

This thesis evolved while I was a research assistant at the Chair of Finance at the

University of Mannheim. It was accepted as dissertation by the Business School of

the University of Mannheim.

This project would not have been possible without the support and help of many.

First of all, I am deeply indebted to my advisor, Prof. em. Dr. Dr. h.c. Wolfgang

Buhler, who initiated this project and who offered me the opportunity to work under

his guidance. He gave his critical advice, his patience, his continuing support and

encouragement throughout my work on this project. Second, I would like to thank

Prof. Dr. Peter Albrecht for co-refereeing this thesis.

Many thanks go to my former colleagues at the Chair of Finance for their

fruitful advice and encouragement, namely to Dr. Jens Daum, Dr. Christoph

Engel, Dr. Sebastian Herzog, Dr. Christoph Heumann, Prof. Dr. Olaf Korn,

Prof. Dr. Christian Koziol, Dr. Jens Muller-Merbach, Dr. Raphael Paschke,

Dr. Marcel Prokopczuk, Dr. Peter Sauerbier, Dr. Antje Schirm, Christian Speck,

Dr. Tim Thabe, Prof. Dr. Monika Trapp, and Dr. Volker Vonhoff, as well as to our

secretary Marion Baierlein. They all contributed to a friendly and stimulating work

environment. In particular, I would like to thank Prof. Dr. Christian Koziol for his

critical and stimulating advice on this topic.

Finally, I would like to thank my family. Their unfailing support and love throughout

the years made this thesis possible.

Mannheim, January 2011 Volker Sygusch

xxi

xxii ACKNOWLEDGEMENTS

Part I

Introduction

1

Chapter 1

The Issue of Pro-Cyclicality

Risk-sensitive capital requirements aim at enforcing appropriate minimum amounts

of capital to absorb losses resulting from credit defaults and other risk sources,

notably market and operational risk. These requirements are considered to be

important as they are supposed to strengthen financial stability. Nevertheless,

practitioners and scholars have worried about negative side effects arising from

capital-based regulation. One such topic concerns the reinforcement of cyclicality

in lending by regulation. These concerns are based upon the notion that regulatory

rules, which become tighter in downturns, restrict the bank’s total lending beyond

what banks would do for reasons of risk management. As a consequence, valuable

projects are not financed during recessions, which may in turn enhance the cyclical

troughs in their depth and length. A situation commonly known as “credit crunch”

or “credit rationing” arises.

In this thesis, we address the question of if and how bank capital regulation may

enhance cyclical patterns in lending. To do this, a bank model is set up, and

different types of capital requirements are considered and compared to the outcomes

if the bank remains unregulated. The mechanisms at work that fuel pro-cyclical

patterns in lending can be also distinguished according to the characteristics of

capital requirements. We add to the literature insofar as the literature about the

pro-cylicality of capital requirements has not been concerned with a bank that

simultaneously takes its leverage, asset risks, probability of bankruptcy, size, and

costs of debt finance into account. Likewise, there is no work of this kind considering

that the deposit volume and the deposit interest rate are based on decisions made

by a risk-averse household and a risk-neutral bank. The deposit interest rate

and the deposit volume reflect the bank’s risk-taking and, if regulation is present,

the regulatory constraint. The impact of capital requirements are analyzed by

3

4 CHAPTER 1. THE ISSUE OF PRO-CYCLICALITY

considering the sensitivity of the lending volume to changes of fundamental economic

variables, called “shocks”, under a VaR approach, under approaches with fixed risk

weights, and under a laissez-faire economy.

The simplest form of capital regulation is to require banks to adhere to a fixed

ratio of equity to assets or deposits. During recessions, credit losses will increase

and diminish the banks’ equity capital positions. Consequently, the volume of the

loan portfolio must shrink in order to attain the regulatory capital ratio once again.

As a consequence, lost loan positions are no longer completely filled. The overall

potential credit capacity of the financial system shrinks.

Under Basel I, capital rules were already more complex than a simple capital ratio:

banks had to meet a weighted capital ratio, i.e. a specific weight was assigned to

each asset position. All weighted asset positions must meet a given relation to

regulatory capital. These weights partly reflected notions of credit risk as some

governmental authorities were given a risk weight of zero and building loans had

preferential weights compared to other loans.1 Therefore, losses could not only

result in decreasing loan portfolio volumes, but also in loan portfolio shifts, here

notably to the disadvantage of plain commercial and retail loans, and in favor of

sovereigns (Haubrich/Wachtel, 1993, p. 3f; Berger/Udell, 1994, p. 586; Furfine, 2001,

p. 34). It has been often attempted to trace back the 1990/91 Credit Crunch to the

introduction of the capital requirements based on the Basel I Accord.

The Basel II rules aimed at determining minimum bank capital in accordance to the

risk taken on the asset side. As recessions are characterized by decreasing prospects

for firms and increasing losses that firms must bear, the firms’ credit-worthiness

shrinks. With increasing credit risk in turn, banks must hold more capital for a

given volume of loans. If, moreover, banks must bear losses themselves, a massive

downturn in bank credit supply will take place according to the critics and sceptics

of risk-sensitive capital requirements. Thus, some people view risk-sensitive capital

requirements as worsening cyclical downturns in credit supply compared to what

risk-insensitive capital requirements may cause.

However, all these criticisms often neglect the fact that lending by its very nature

may be a cyclical business. In the next sections of this chapter, we present the

literature that is concerned with the issue of if and how capital regulation may

result in pro-cyclical patterns in lending.2

1Dewatripont/Tirole (1994, Ch. 3), provide an account of the implemented rules based onthe Basel I Accord, particularly for the EU. In Germany, these rules were implemented as the‘Grundsatz I’ of which §13 listed all eligible risk weights.

2For different reviews of literature on banking regulation, in particular capital requirements, we

5

To start with, Section 1.2.2 presents theories of how lending and business cycles

may interact in the absence of regulation. Section 1.2 reviews the introduction

of the Basel I Accord and the 1990/91 Credit Crunch. The empirical literature

that asks if and to what extent the introduction of Basel I was responsible for the

downturn in granting corporate loans, is discussed. Section 1.2.2 presents theoretical

arguments in favor of pro-cyclical effects through capital requirements. Section 1.3

is devoted to the Basel II Accord. Its basic mechanisms are presented (Section

1.3.1). The introduction of further amendments, commonly known as Basel III are

discussed. Section 1.3.2 reviews the literature that is concerned with the cyclicality

of the required minimum capital. Section 1.3.3 discusses the role of capital buffers,

i.e. the excess capital that is held by commercial banks beyond what is required

by regulators. Section 1.3.4 highlights the impact that the type of the loan rating

system has on the cyclicality of capital requirements. Section 1.4 points out that

there are also other regulatory norms that may fuel pro-cyclicality. Finally, Section

1.5 discusses the impact capital adequacy rules may have on the risk-taking of banks.

The next chapter, Chapter 2, outlines the framework of our analysis. It refers to the

types of shocks considered, our notion of pro-cyclicality, and to the basic framework.

The following two parts, Part II and III, are the core of this thesis. Two distinct

versions of the basic model described in Chapter 2 are analyzed under different

regulatory regimes. The impacts of shocks under these regimes are assessed

using comparative static analyses which are performed numerically, except for few

instances of equilibria. In Part II, the bank grants loans to two firms which run

different, potentially correlated projects whose outcomes are Bernoulli distributed.

The analysis is restricted to one period. Credit risk is either regulated by fixed risk

weights or by a VaR approach. Part II consists of two chapters.

In Chapter 3, the model set-up is explained and some theoretical implications are

derived. In Chapter 4, the question of pro-cyclicality resulting from capital adequacy

rules is numerically discussed. Part II also addresses issues of risk-taking.

In Part III, the bank faces many firms whose different Bernoulli-distributed projects

are weakly correlated and whose projects can be lumped together into two distinct

sectors. In the aggregate, the bank faces normally distributed loan-portfolio returns

on which the bank’s and the household’s decisions are based (Ch. 5). Chapter 6

is devoted to the analysis of the one-period model. In Chapter 7, the analysis is

extended to two periods. Both fixed risk weights and a VaR approach are considered

refer to Bhattacharya/Boot/Thakor (1998), Santos (2000), Stolz (2002), Van Hoose (2007,2008),and Borio/Zhu (2008).


for the one-period analysis as regulatory capital requirements, whereas the analysis

of the two-period model is restricted to a comparison of the laissez-faire equilibrium

to the equilibrium under the VaR approach. The final part, Part IV, contains

conclusions. Derivations of theoretical results can be found in the Appendix.

A main result is that the effect of a given shock depends on the type of the shock.

Equity shocks, i.e. changes in bank equity as a result of gains or losses, lead to

pro-cyclical effects on lending. Expectation shocks, i.e. changes of distribution

parameters or of borrowers’ productivity, are dampened by regulation. Thus,

regulation may exacerbate downturns after losses have been realized, whereas

regulation constrains bank lending if more favorable outcomes are expected, thus

hampering economic recovery. Moreover, there is evidence of counter-cyclical effects

through regulation on the level of single loan volumes. These observations can

be made regardless of the risk sensitivity of the respective capital adequacy rule.

Interestingly, a regulated bank may grant higher loan volumes to less risky firms

if risk-sensitive capital requirements are binding than it would do under fixed

requirements or under a laissez-faire regime.

The two-period model supports these findings. In particular, there are no signs

that a financial accelerator based on a propagation mechanism via the bank’s equity

exists. That is, the sensitivities of the expected total loan volume in the second

period with respect to expected equity which prevails after the first period are always

the same size as they are in the one-period model with respect to the bank’s initial

equity.

Furthermore, this thesis shows that enhanced risk-taking may occur under capital

regulation in the absence of other regulatory requirements, such as deposit insurance.

Rather, a flat capital requirement alone induces the bank to take more risk than it

does under any other capital adequacy regime considered in this thesis (i.e. different

risk weights, VaR approach) and than under laissez-faire.

1.1 The Pro-Cyclicality of Lending

As the business cycle is characterized by up- and downturns in aggregate output,

it is of interest of how firms’ access to finance and firms’ financing demands are

affected. The standard IS/LM model is frequently used in first place to analyze of

how the goods market is affected by financing conditions, notably by the interest

rate and by money supply. The standard IS/LM model does not address monetary

transmission via the bank loans, however.

1.1. THE PRO-CYCLICALITY OF LENDING 7

Bernanke/Binder (1988) were the first who studied the role of commercial lending

in the IS/LM-framework. The conventional LM curve, that only recognizes money

and bonds as financial assets, is extended by (commercial) loans. As a consequence,

(government) bonds and (commercial) loans are no longer perfect substitutes. Ber-

nanke/Blinder derive loan supply from a generic bank balance sheet and the loan

demand relation is set ad hoc. The standard IS/LM-model thus becomes a limit

case of their model.

If loans and bonds are perfect substitutes, the curve representing the equilibria on

the goods and credit market reduces to the familiar IS-curve. If money and bonds

are perfect substitutes, the economy slips into the well-known liquidity trap (Ber-

nanke/Blinder 1988, p. 436). In between, there is now a role for the credit market

to affect the goods market and total equilibrium. In particular, a positive credit

supply shock raises bond interest rates and aggregate output. Furthermore, reserve

requirements may take more expansionary effects than in the standard IS/LM model.

Bernanke/Gertler/Gilchrist (1999) refine the notion of how credit and credit

availability may cause macroeconomic up- and downturns. They incorporate

credit market imperfections into a dynamic, general equilibrium New Keynesian

model. Firms may borrow funds, but at a rate that is higher than their

internal cost of capital. The spread, termed external finance premium, is due

to agency problems and depends inversely on the firm’s net worth, whereas the

latter is as such pro-cyclical. These two ingredients form the financial accelerator

(Bernanke/Gertler/Gilchrist, 1996, p. 4). Contrasting this model with conventional

New Keynesian frameworks serves to compare the impacts that the firms’ net

worths have on transmitting shocks. Numerical studies illustrate that the thus

implied financial accelerator propagates and amplifies shocks. This view, however,

stands in contrast to the work of Bernanke/Blinder (1988) and Blum/Hellwig

(1995)3 insofar as the latter consider amplifications via the banks’ ability of

granting loans. However, the existence of the financial accelerator in the sense

that lending amplifies macroeconomic shocks cannot be confirmed for Germany

(Eickmeier/Hofmann/Worms, 2006).

One alternative explanation of the cyclicality of lending is the so-called “institutional

memory hypothesis” which has been put forward by Berger/Udell (2004). This

hypothesis says that the loan officers’ abilities to judge loans according to their

default risks deteriorate after the last bust as time has passed. As a result, credit

standards are eased with upswings, notably by decreasing premia. Berger/Udell

3Their work is discussed in Section 1.2.2, p. 10.


(2004) analyze data on US banks to support this thesis. They find a statistically and

economically significant relation between loan growth and the time elapsed since the

last busts. As they also control for other demand and supply factors, they regard

this significant relation as sufficiently supportive for their hypothesis. However,

as also Berger/Udell (2004) note, the institutional memory hypothesis could also

simultaneously show up with other phenomena. Furthermore, it could be criticized

as a truism. In this vein, without recurring on this hypothesis, Ayuso/Perez/Saurına

(2004, p. 261) note that banks “might tend to behave during a boom as if it were

to last for ever.”

1.2 The Basel I Accord and Pro-Cyclicality

1.2.1 The Introduction of Basel I and the 1990/91 Credit

Crunch

At the beginning of the nineties the Basel-I-rules were introduced and some further

regulatory changes came into force in the USA. More specifically, those requirements

were partially introduced in 1990 and they finally went into full effect in 19924

Moreover, a minimum capital ratio based on the total, non-weighted asset volume

was required from 1990 on (cf. Berger/Udell, 1994, p. 585, and Berger/Herring/Sze-

go, 1995, p. 403).

During this period, the aggregate loan volume remained at a constant level

in the USA whereas the volume of government securities held by commercial

banks increased (Haubrich/Wachtel, 1993, p. 3f; Berger/Udell, 1994, p. 586).

In turn, the total of commercial and industrial loans outstanding strongly went

down from 1990 on, and the bank capital held increased (Haubrich/Wachtel,

1993, p. 3f; Berger/Udell, 1994, p. 586; Furfine, 2001, p. 34). In particular,

Berger/Herring/Szego (1995, p. 402f) report that US bank capital ratios raised from

6.21% at the end of 1989 to 8.01% at the end of 1993. It was widely attempted to

regard the new regulations as cause for the observed changes in the banks’ portfolios.

However, not all academics were convinced by the view that regulation caused a

credit crunch or even worsened the macro-economic decline. Yet, the obvious decline

4Cf. Avery/Berger (1991, p. 862-864) concerning the prompt-corrective action plan, andBerger/Udell, 1994, p. 585 concerning the introduction of Basel I in the USA. According to Rochet(1992, p. 1137) and Freixas/Rochet (1997, p. 239), the capital requirements based on the Basel IAccord went into effect in 1993 for all commercial banks in the EEC.

1.2. THE BASEL I ACCORD AND PRO-CYCLICALITY 9

in commercial and industrial loans on the one hand and the rise of government

securities on banks’ balance sheets on the other, fueled concerns that capital

requirements could worsen recessions. In other words, the fear of “pro-cyclical”

effects on lending by capital regulation was born. Furthermore, the observed

portfolio shifts in favor of default-risk free securities added to the debate about

the relation between risk-taking and minimum capital requirements which will be

addressed later in this chapter.

The hypothesis that the introduction of the Basel I capital requirements

caused the credit crunch in the US is supported by Haubrich/Wachtel (1993),

Brinkmann/Horvitz (1995), Shrieves/Dahl (1995), and Jackson et al. (1999),

amongst others. In particular, Jackson et al. (1999) claim that Basel I possibly

contributed to constrained lending in Japan, too, whereas they confirm for the

remaining G-10 countries that the introduction of Basel I seems to have induced

weakly capitalized banks to increase capital ratios only. Similarly, Wagster (1999)

finds evidence of constrained lending through Basel I for the US and Canada at the

beginning of the nineties, but does not find it for the UK, Germany, and Japan.

Aggarwal/Jacques (1998) attribute the increasing bank capital ratios in the USA

during the years 1991 to 1993 to the PCA plan. This view is also held by Furfine

(2001, p. 51). Although he admits that an increase in risk-based capital requirements

could have added to the credit crunch, this increase could not fully explain it. Ber-

ger/Herring/Szego (1995) loosely ascribe the increase of US bank capital ratios to

the introduction of the new capital requirements, the PCA plan, and of the risk-

based deposit insurance premia. Peek/Rosengreen (1995) concentrate their study on

New England and find evidence that banks were strongly constrained by declining

capital, calling this phenomenon “capital crunch”. As data on loans is too scarce,

they cannot confirm the existence of a credit crunch, i.e. decline in bank loan supply.

The view that the introduction of risk-based capital requirements and other

regulatory measures during 1990 to 1992 caused the credit crunch is not unanimously

shared amongst academics, however. Berger/Udell (1994, p. 625) give evidence that

alternative supply side mechanisms “are somewhat more consistent with the data”

but their “quantitative effects are not substantial.” But they also believe that non-

risk related credit crunch hypotheses, as regulatory pressure, could have played a

role. Bernanke/Lown (1991) consider a weak loan demand as main cause for the

slowdown in lending. However, they do not deny that weak bank capital bases

have impaired banks’ lending potential. A full account of the various hypotheses

considered to have caused the decline in loans at the beginning of the 1990s in the


US is provided by Berger/Udell (1994, pp. 586-588).

1.2.2 Theoretical Research on the Pro-Cyclicality of Basel I

Blum/Hellwig (1995) were the first5 to deal with the issue of pro-cyclicality of

capital requirements. The implications of a flat regulatory capital-to-asset ratio

are analyzed within an IS/LM-framework. To be able to illustrate its potential

propagating effects, they introduce a stylized banking sector. Blum/Hellwig (1995)

thus distinguish in contrast to standard IS/LM analysis between commercial loans,

government securities, and demand deposits where the latter is determined by the

money demand. The banking sector comprises equity and demand deposits on the

liability side, and commercial loans and bonds on the asset side. Credit risk and

other types of risks that prevail in banking are neglected.

Given a real shock to aggregate demand, regulation causes a raise in the aggregate

demand multiplier that is higher than the raise in the aggregate demand multiplier

in the otherwise same economy where the banking sector is not regulated. Put

differently, the sensitivities of equilibrium prices and output relative to shocks

increase as well. Thus, regulation enforces pro-cyclicality. But their model cannot

address the question of how much risk the banking sector takes after a shock has

occurred and to what extent shocks have an impact on the risk positions of firms

and depositors.

Thakor (1996) adds to the view that regulatory measures can cause a decline in loans

by proving that increased capital requirements lead to increasing credit rationing

in the sense that the borrower’s probability of being denied for credit goes up.

The argument builds on asymmetric information and the bank’s cost of screening.

Furthermore, Thakor (1996) asks if monetary policies lose their effectiveness under

a regime of minimum capital requirements. He conjectures that monetary policy

may have opposite effects on aggregate lending depending on how these impulses

affect the term structure of interest rates.

Bliss/Kaufman (2003) show by considering a simplified bank balance sheet that the

injection of an additional amount of reserves beyond what is initially required leads

to a lower increment in earning assets if capital requirements are binding than it

is the case without capital requirements. As a consequence, a considerably large

amount of total reserves is held in excess given binding capital requirements. Thus,

these excess reserves, that would allow for an appropriate increase in deposits, and

5Cf. Freixas/Rochet, 1997, p. 274.

1.3. THE BASEL II ACCORD AND PRO-CYCLICALITY 11

hence in earning assets, can only be used to expand the volume of earning assets if

new equity capital is raised. But in recessions, this is unlikely to occur as it becomes

too costly. So, capital requirements reinforce cyclical downturns regardless of their

risk-sensitiveness.

1.3 The Basel II Accord and Pro-Cyclicality

1.3.1 The Basel II Accord

The Basel II Accord (BCBS 2004), which has taken effect in Germany on the 1st of

January 2008 (Deutsche Bundesbank, 2009, p. 55), offers the Standardized Approach

and the “Internal-Ratings-Based Approach” (IRB Approach) for determining the

required minimum capital for credit risk.6

In principle, the Standardized Approach assigns a risk weight to any claim according

to the economic sector to which the debtor belongs and according to the debtor’s

credit risk. The credit risk appraisal should be reflected in a credit grade given by an

external credit assessment institution (BCBS, 2004, Art. 52). Risk weights usually

comprise several credit grades given a class of debtors. There are risk weights for

claims on sovereigns and central banks (BCBS, 2004, Art. 53, 55), banks (ibid.,

Art. 63), and corporates (Art. 66). Interestingly, claims secured by residential

property are assigned a weight of 35% (Art. 72f), whereas a weight of 100% is

required concerning commercial real estate because of the turbulence in the 1980ies

and 1990ies (Art. 74). The Standardized Approach can be seen as a refinement of

the Basel I Accord that merely distinguished claims by their sector-specific origin

or by the collateral pledged.7 Consequently, risk sensitivity of capital requirements

has increased from the Basel I to the Basel II Accord.

Risk sensitivity increases even further if the Basel I rules are compared to the IRB

Approach. The risk weights of the IRB Approach are derived from a credit risk model

where the returns of claims follow a one-factor model with normally distributed

factor loadings and where the claims are fully granular. Gordy (2003) characterizes

6We do not consider minimum capital requirements for securitization which follow differentrules (cf. BCBS, 2004, pp. 113-136). In the European Union, the directives 2006/48/EC and2006/49/EC build the legal basis for national law-setting. In Germany, the law concerning thenew capital requirements based on the Basel II Accord (“Solvabilitatsverordnung”) came intoforce on the 1st of January 2007 allowing for a one-year transition period. In this thesis, we willexclusively refer to the document launched in June 2004 by the Basel Committee on BankingSupervision (BCBS, 2004).

7For Germany, we refer to §13 Grundsatz I, listing all then eligible risk weights.


the assumptions that the portfolio be fully granular and that there be at most a

single systematic risk factor as necessary and, with a few regularity conditions, as

sufficient to obtain the portfolio-invariant additivity of risk weights, laying thus

the theoretical foundations for the IRB Approach. Portfolio-invariance means that

the capital charge of each asset item does not depend on the characteristics of

the associated portfolio.8 Given the underlying model, the IRB risk weights yield

the 0.1%-quantile of a notional portfolio’s credit loss distribution (BCBS, 2004,

Art. 272). Hence, required minimum capital in the IRB approach can be understood

as the capital level that absorbs all losses with 99.9% probability.

Thus, many theoretical papers, which model the portfolios’ returns differently

from what the IRB Approach assumes, represent this approach by requesting the

bank to keep a given confidence level based on a VaR approach for economic

capital. Models using VaR to represent the Basel II regime include Buhler/Koziol

(2005), Dangl/Lehar (2004), Danıelsson/Shin/Zigrand (2004), Estrella (2004), and

Danıelsson/Zigrand (2008). We follow this modeling approach in our analysis, too.

As the increased risk sensitivity in capital requirements is obvious, much research

has been devoted to the cyclicality of minimum required capital compared to the

Basel I regime. This literature will be reviewed and discussed in Section 1.3.2.

Section 1.3.3 reviews the work concerned with capital buffers, i.e. the amount of

capital held above what is required by regulation. Section 1.3.4 is devoted to the

role of loan rating systems which has attracted a lot of attention since banks must

estimate the probability of default (PD) in the IRB Approach and may also estimate

other inputs after approval (BCBS, 2004, Art. 245-248).9 Because these estimates

are sensitive to the credit cycle in one way or the other, these estimates may be

an additional source of cyclicality. Section 1.3.5 presents further literature that is

concerned with pro-cyclicality due to risk-sensitive capital requirements.

As a response to the recent subprime crisis that in effect infected many other financial

markets, amendments to the Basel II framework have been discussed in depth since

2009. These amendments have been known as Basel III and comprise a bundle of

measures (BCBS 2010). These new rules are concerned with the quality and size of

the bank’s equity capital, the total leverage, and liquidity. The new rules will be

phased in from 1st January 2013 on and shall take full effect from 2019 on. In times

8Outlines of the specific model and a derivation of the thus implied risk weight can also befound in Vasicek (1991), Schonbucher (2000), and Hartmann-Wendels (2003, pp. 117-120).

9These two options are called “foundation approach” and “advanced approach”. The foundationapproach requires that banks only estimate the default probabilities, whereas under the advancedapproach banks have to estimate all the IRB-model’s input parameters.


of stress, banks will be allowed to draw on the “conservation buffer” which will be

introduced from 2016 on. The conservation buffer shall thus address the problem

that banks reach the regulatory minimum amount of equity capital especially in

adverse economic situations in which new equity cannot be raised. This option

to absorb losses, however, will be linked to tighter regulatory constraints on the

distribution of earnings (BCBS, 2010). Another lesson from the recent financial

crisis has led to stricter liquidity rules.10

Most of importance in respect to this thesis is the attempt to establish an anti-

cyclical buffer (cf. BCBS 2010). The buffer will account for 0 to 2.5 percentage

points of the capital ratio. It is supposed to curtail credit growth in expansive

economic phases. So the build-up of excessive risk shall be contained. Consequently,

this anti-cyclical buffer will not change results concerning cyclicality compared to

the current Basel II framework when downturns are considered. As the results of

this thesis suggest that capital requirements already lead to lower loan volumes and

in particular to lower credit growth than it is the case in a regime of laissez-faire,

we shall not expect different qualitative results concerning upturns, either.

1.3.2 Pro-Cyclicality of Required Minimum Capital

The empirical research on the 1990/91 credit crunch has strengthened the

view that bank capital and bank capital requirements are key drivers of total

lending. Consequently, many scholars feared that an increased cyclicality in capital

requirements by the Basel II Accord could amplify fluctuations in lending and thus

in aggregate output.

For example, Bikker/Metzemakers (2007) find, based on bank data over OECD

countries, that regulatory capital over risk weighted assets is pro-cyclical whereas

the ratio of capital over risky assets is mostly not. Particularly, Bikker/Metzemakers

(2007) and Bouvatier/Lepetit (2008) detect a tendency that riskier (and also smaller)

banks maintain relatively low ratios. This relation implies that the regulatory capital

of those banks may rather infringe regulatory thresholds if these thresholds depend

on risk.

The works of Carpenter/Whitesell/Zakrajsek (2001), Ervin/Wilde (2001), Sego-

viano/Lowe (2002), Goodhart/Hofmann/Segoviano (2004), Kashyap/Stein (2004),

Illing/Paulin (2005), and Gordy/Howells (2006) determine what amount of capital

10For an account of the crisis cf. Calomiris (2008) and Hellwig (2008), amongst others. Withrespect to the perceived phenomenon of (wholesale) bank-runs, we refer to Gorton (2008, 2009).


would have been required in the recent past for notional loan portfolios, that they

deem representative, if Basel II had been already in place.

Their findings can be summarized as follows:

Segoviano/Lowe (2002) and Goodhart/Hofmann/Segoviano (2004) find strong

amplifications in required minimum capital for Mexican banks in the 1990ies.

Their simulations of the loan portfolios, that are based on Mexican, Norwegian,

and US-American data each, show that required minimum capital is more

cyclical under the IRB Approach than under the Standardized Approach

(Goodhart/Hofmann/Segoviano, 2004). They observe this enhanced cyclicality even

though they feed the IRB formulæ with Moody’s data which are regarded as through-

the-cycle estimates.

Illing/Paulin (2005) confirm for Canadian banks for the years 1984 to 2003 that

required capital would have become more volatile under Basel II. Kashyap/Stein

(2004) detect that Basel II rules might have raised required minimum capital by

2% to 160% compared to Basel I. Ervin/Wilde (2001) determine that bank capital

would have been required to increase by 3% under the Basel I and by 20% under

Basel II for US-loan portfolios during 1990. The increase in required capital from

Basel I to Basel II is based on credit migrations.

Also Hofmann’s (2005) simulation study suggests that the internal IRB approach

enhances the cyclicality of lending compared to the Basel-I-framework, but that

both regulatory regimes dampen this cyclicality compared to the economic capital

approach pursued by the otherwise unregulated bank. The set-up is kept simple:

the bank’s equity is exogenously fixed, debt-finance not considered, and the loan

portfolio is homogeneous. The study can only account for changes in the total loan

volume that are based on cyclical changes of capital requirements. Credit risk is

based on a one-factor model. Cyclical swings in credit risk are exclusively given by

increasing and decreasing default barriers of the obligors in the bank’s loan portfolio.

The fluctuation in the default barrier is in turn given by a stylized, sine-shaped

business-cycle model and by the evolution of total lending.

To sum up, the studies of Ervin/Wilde (2001), Segoviano/Lowe (2002),

Goodhart/Hofmann/Segoviano (2004), Kashyap/Stein (2004), Hofmann (2005), and

Illing/Paulin (2005) give evidence that the switch from Basel I to Basel II implies

an increase in the amount and in the cyclicality of required capital.

In contrast, Carpenter/Whitesell/Zakrajsek (2001) do not find any substantial

increased cyclicality in bank capital requirements under the Standardized Approach


of Basel II compared to those of the preceding risk weighting rules. Their analysis

is based on a study of a representative loan portfolio during the years from 1970

to 2000. Gordy/Howells (2006) put the studies suggesting increasing cyclicality

through Basel II into perspective by illustrating how different re-investment rules

after loan defaults affect the cyclicality of capital. Their study is based on a dynamic

simulation setting. In particular, they can show that re-investment rules may have

a larger impact on the cyclicality of required capital than capital adequacy rules do.

All these studies consider the cyclicality of bank capital exclusively as a matter of the

cyclicality of required capital. As required capital results from the composition of

the bank’s loan portfolio, the lending policy is crucial for the total result. This strand

of literature emanates from historic, average loan portfolios and from exogenously

fixed portfolio strategies.

The appeal of the literature that is concerned with the cyclicality of minimum capital

requirements is two-fold: first, a considerable number of these studies have a strong

relation to real-world bank portfolios. Second, this literature considers the evolution

of capital through time, often for the full length of a business cycle.

As shortcomings, three important points can be listed. First, these studies neglect

potential banks’ internal capital targets. Second, they neglect potential endogenous

reactions to a new set of capital rules which seems to be important in light of the

theoretical literature on risk-taking as well as in light of the empirical studies on

the 1990/91 credit crunch. In contrast, the bank reacts differently given the various

regulatory regimes and must consider the re-financing costs associated with its risk-

taking in this thesis. Third, as in Section 1.3.3 discussed, banks hold capital buffers

on top of what is required. Therefore, their behavior and their potential influence

on capital cyclicality shall be discussed.

Estrella (2004) is also concerned with the cyclicality of bank capital but some of

these drawbacks are addressed as he analyzes the bank’s optimal capital choice with

and without regulation in a dynamic, infinite-horizon model.

The bank’s choice of its level of equity is associated with costs. These costs are the

cost of holding capital, of failure and of capital adjustment. Both bank loans and

deposits are given exogenously.

If adjustment costs are neglected, the optimally chosen level of capital is identical to

the economic capital calculated by the VaR approach at an endogenously determined

confidence level. This characterization allows to decide whether a confidence level

set by regulation is binding or not.


If adjustment costs are introduced and if losses follow a stylized business cycle,

the optimal level of capital of the unregulated bank is negatively correlated with

the VaR-restricted capital level if separately determined period-by-period by the

regulator. The reason is that the latter neglects adjustment costs such that capital

is sooner raised after a shock in loan losses than the bank would optimally do without

regulation.

Furthermore, the VaR constraint is positively related to external capital flows.

Hence, regulation is expected to be binding under adverse economic situations,

giving rise to the concern that regulation amplifies recessions.

Since there is no endogenous trade-off between raising new external capital and

granting loans, the issue of pro-cyclicality of loan volumes cannot be directly

addressed. Arguments can only be indirectly run via capital decisions and

restrictions. The model cannot address the question of risk-taking under regulation,

either, because the bank is only concerned with optimally adjusting its equity

capital. Though adjustment costs are considered in this study, too, these costs

are symmetrical for both up- and downswings so that equity can be raised during

recessions at unchanged conditions.

Yet, it is questionable whether we should be concerned with cyclicality in capital

at all: bank capital cannot be adjusted easily, especially not in times of increasing

numbers of loan defaults. Rather, bank capital is fixed, only changing with losses

and retained earnings. What must be adjusted, given the credit risk perceived and

the capital given, is the bank’s total loan volume. Hence, the lending volume should

be the variable of first interest.

1.3.3 Pro-Cyclicality and Capital Buffers

In reality, banks hold capital beyond what regulation requires. This excess capital

is usually referred to as capital buffer.

Banks hold capital buffers to reduce the cost of infringing capital requirements.

The cost of infringing capital requirements is associated with the loss of the bank’s

charter or franchise value, which becomes the higher, the higher future expected

returns are. According to Milne/Whalley (2001), increased recapitalization costs or

a higher auditing frequency can further raise the bank’s internal target of capital.

Alfon/Argimon/Bascunana-Ambros (2004, p. 20) confirm that UK-based banks and

thrifts hold capital buffers in order to avoid costly recapitalization when needed

most. Jackson/Perraudin/Saporta (2002) explain capital buffers empirically by the


improved access to the swap market in terms of contract conditions. Lindquist

(2004) finds that Norwegian banks merely hold capital buffers to signal solvency to

the market and to the regulator. Moreover, regulators suppose banks to hold capital

buffers as well (BCBS, 2004, Art. 757f).

The existence of capital buffers raises the question of whether research on pro-

cyclicality of minimum capital requirements and lending may become pointless. In

particular, it could be claimed that it becomes dispensable to compare capital and

lending under the Basel I regime with those under the Basel II regime. This view is

held by Bikker/Metzemakers (2007) who do not fear pro-cyclical effects on lending

with the introduction of Basel II despite the pro-cyclicality of capital ratio (cf. p. 13

of his thesis) since they can show that most banks hold significant capital buffers

enabling them to smooth fluctuations in realized and expected losses.

The notion that capital buffers might absorb some of the cyclicality inherent in

minimum capital requirements is sustained by the empirical magnitudes that Er-

vin/Wilde (2001) and Furfine (2001) report. According to Ervin/Wilde the IRB

weights proposed at that time resulted in half of what had been required under the

Basel I Accord whereas Furfine (2001, p. 52) points out, based on Carey’s (1998)

data, that roughly two thirds of bank loan portfolio are rated BBB or BB and thus

are not subject to changes in capital requirements if the Standardized Approach is

used. Jackson/Perraudin/Saporta (2002) estimate that most banks of their sample11

aim at a survival probability higher than 99.9% whereas the Basel I framework only

implies survival probabilities ranging from 99.0% to 99.9%. In particular, these

banks should have met ceteris paribus the minimum capital requirements after the

adoption of the Basel II rules.

Peura/Jokivuolle (2004) calculate the capital charges according to Basel I and Basel

II for high-quality and average-quality loan portfolios.12 Furthermore, they add

a capital buffer such that the regulatory capital amount is not infringed by a

probability of 99%. This probability is referred to as “the confidence level applied to

regulatory adequacy” (ibid., p. 1809). This strategy of holding a buffer is compared

to holding economic capital such that solvency is kept by a probability of 99.95%

which is in line with the empirical findings of Jackson/Perraudin/Saporta (2002). In

sum, the simulation study of Peura/Jokivuolle (2004) results in capital buffers that

lead to a less cyclical total capital amount than it is the case if the bank seeks to keep

11Their data is based on a Federal Reserve Board survey and by Deutsche Bank’s annual report.The data comprises the distributions of credit quality in banks’ loan portfolios.

12The percentages of credit grades are along the lines of Gordy’s (1998) data on US loanportfolios.


a given solvency probability. These results hold for both loan portfolio qualities and

for both regulatory regimes, i.e. Basel I and II. Interestingly, capital buffers become

the higher relative to required capital, the higher the portfolio’s loan quality is as

the potential for down-side movements grows.

Gambacorta/Mistrulli (2004) give empirical evidence of the notion that capital

buffers smooth the cyclicality in lending as they show for Italian banks that the

higher capital buffers are, the less pro-cyclical lending becomes after GDP-shocks

have realized.

Jacques (2005) and Heid (2007) illustrate the role of capital buffers concerning the

pro-cyclicality of risk-based capital requirements in a framework that is based on

the model used by Blum/Hellwig (1995). Jacques (2005) and Heid (2007) emphasize

that pro-cyclicality of risk-based capital requirements can only be mitigated if, at

least, capital buffers strictly increase in aggregate output and thus move in opposite

direction to what has been observed under the Basel I regime.13 The results of

Repullo/Suarez (2008) show that such a behavior of capital buffers under Basel II

may prevail.

Both, in Jacques’ (2005) and Heid’s (2007) models, the single, portfolio-wide risk

weight depends on aggregate output. Jacques (2005) shows that ratings migrations

unambiguously can add to increased cyclicality in lending despite of capital buffers

held. Heid (2007) contributes to the literature with two findings: first, capital buffers

may absorb the cyclicality of required capital, thus mitigating the overall effect on

the capital ratio. Second, the capital buffer under risk-sensitive capital requirements

decreases in downturns.

In Heid’s (2007) model, the representative bank maximizes its expected profits given

its funding constraint while keeping a given probability not to infringe the required

regulatory equity amount. The latter constraint results in the capital buffer. Its

dependence on aggregate output is given by the assumptions that underpin the

macroeconomic framework used. A shortcoming of the study of Jacques (2005) is,

however, that the capital buffer and its relation to the business cycle in terms of

aggregate output is exogenous.

Kajuth (2008) demonstrates in a micro-economic model of a banking sector that

the degree of pro-cyclicality may well depend on the degree of capitalization and

13A lot of empirical evidence on the counter-cyclicality of capital buffers with respect to GDPhas been gathered, including Ayuso/Perez/Saurına (2004) for Spanish banks, Bikker/Metzemakers(2007) for OECD-countries based banks, Jokipii/Milne (2006) for most types of EU-25 basedbanks, Lindquist (2004) for Norwegian banks, and Stolz/Wedow (2005) for German savings andcooperative banks.


on the behavior of the interbank lending rate toward aggregate risk. The degree

of capitalization reflects the distribution of capital buffers and, in particular, also

entails a subset of banks that face binding capital requirements. Similarly to Jacques

(2005) and Heid (2007), capital requirements are inversely related to aggregate

output. Given intermediate levels of capitalization, and a low elasticity of the

interbank rate with respect to aggregate output, regulation strongly affects lending

in a pro-cyclical way. For high elasticities the opposite is true.

Zhu (2008) illustrates within a dynamic bank model that capital buffers absorb some

of the cyclicality of minimum capital requirements. But the calibrated model also

shows that banks that are not effectively constrained by capital requirements may

even exhibit the strongest cyclical swings in lending as they can freely engage in

lending during upswings. The lending cycle of effectively constrained banks, though

holding a buffer, are less pronounced.

As in Zhu’s (2008) analysis, banks, that are constrained by regulation, hold a capital

buffer on what is required, it remains unclear to what extent regulation may actually

affect lending in a pro-cyclical way in his model. In general, it is not clear where the

benchmark must be set so that a comparison of a constrained and an unconstrained

banks becomes meaningful in reality. The work of Zhu (2008) is close to this thesis

insofar as the effects of capital requirements on the lending volume dependent on

the state of the economy are analyzed. In contrast to our approach, depositors are

assumed to be risk-neutral and there is one single bank asset only of which the

volume strictly decreases in the total loan volume.

Over and above, we take the view that holding capital in excess of what regulation

requires does not extinct the problem of potential pro-cyclical effects on lending for

two reasons: first, if capital buffers are held to keep the probability of being closed

by regulatory authorities small, and if these buffers are either absolutely fixed or

are a fixed portion of required capital, pro-cyclicality remains an important issue

to investigate. Second, as Repullo/Suarez (2008, p. 35) point out, extrapolating

the behavior of the capital buffers under the Basel I to the Basel II regime may

be misleading as the new requirements are supposed to have a different impact on

bank behavior in general than the Basel I requirements just had. Thus, simply

extrapolation past behavior does not withstand the Lucas critique according to

Repullo/Suarez (2008).

In particular, they show within a two-period framework that capital buffers become

pro-cyclical under risk-sensitive capital requirements. In contrast, capital buffers

are counter-cyclical if a Basel-I-style framework is in place.


The latter result is in line with empirical findings. Different empirical studies show

in deed that the capital buffers held by banks have been inversely linked to GDP and

have been thus counter-cyclical.14 Yet, it is still questionable to what extent total

capital actually absorbs the cyclicality of required capital. Alfon/Argimon/Bas-

cunana-Ambros (2004, p. 29) observe for UK-based banks and thrift institutions

that at most half of the changes in capital requirements have been absorbed by the

buffers held in the period from 1997 to 2002 under the Basel I regime.

1.3.4 Pro-Cyclicality and Loan-Rating Systems

Pro-cyclicality in required capital or lending might not be due to the risk sensitivity

of the Basel II Accord alone, but also due to the mode of how risk is measured.

Concerning risk measurement techniques, two main classifications have emerged:

through-the-cycle (TTC) and point-in-time (PIT) estimates. TTC estimates are

based on data that comprise at least the full length of an economic cycle, as its

name suggests. PIT estimates, however, rest upon data of short time periods. PIT

estimates may serve as basis for the calculations of TTC. Consequently, PIT are

more prone to changes with the economic cycle going ahead whereas TTC estimates

are rather constant unless the reference entity goes into bankruptcy or faces any

other severe idiosyncratic shock (e.g. Amato/Furfine, 2004).

Additionally to estimated default probabilities, estimated loss given default rates

may result in a further source of cyclicality as the study of Altman/Resti/Sironi

(2002) suggests. They detect a negative relation between recovery rates and default

probabilities and confirm by simulations that considering this correlation in the IRB

approach enhances pro-cyclicality.

All in all, it seems intuitive that IRB-based capital requirements based on PIT

estimates will fluctuate stronger than if the IRB formulæ are fed by TTC estimates.

In principle, this notion carries over to the Standardized Approach, where risk

weights only change in a discrete manner. This intuition has been confirmed by

empirical studies and simulations.

Concerning the Spanish mortgage market, Saurına/Trucharte (2007) conclude that

TTC-ratings could much alleviate pro-cyclicality compared to PIT-ratings used in

the IRB formulæ. In the same vein, Illing/Paulin (2005) confirm that the volatility of

14Cf. Ayuso/Perez/Saurına (2004) for Spanish banks, Bikker/Metzemakers (2004) for OECDcountries, Jokipii/Milne (2006) for pan-European samples, Lindquist (2004) for Norway, andHeid/Porath/Stolz (2004) as well as Stolz/Wedow (2005) for German savings and cooperativebanks.


required capital is higher under the use of PIT estimates, derived from bond credit

spreads, than under the use of TTC ratings. For low-rated assets this problem

may worsen since the respective volatility increases with decreasing loan quality.

Likewise, Kashyap/Stein (2004) detect the highest and the lowest increments in

capital requirements under the KMV model which is supposed to deliver PIT

estimates, but not under the credit ratings provided by S&P, which are regarded

as TTC estimates. Also Goodhart/Hofmann/Segoviano (2004), who judge the IRB

foundations-approach as highly pro-cyclical according to their simulation results,

emphasize that capital requirements could become even more pro-cyclical in reality

as banks regard their internal ratings as PIT whereas their simulation study and

their calculations of default probabilities are based on Moody’s TTC estimates.

Catarineu-Rabell/Jackson/Tsomocos (2005) ask within a general-equilibrium set

up which rating system banks optimally choose if they can select between a pro-

cyclical, counter-cyclical, and a cycle-neutral risk weight for corporate loans. More

specifically, a pro-cyclical risk weight strictly decreases as the expected recovery

rate increases whereas the opposite holds for a counter-cyclical risk weight. This

assumption can be thought of as if expected recovery is assigned to applicable

probability-of-default bands in different ways. The cycle-neutral risk weight remains

constant throughout the cycle. Capital requirements are assumed to be binding at

the end-date.

If banks are free to choose, they opt for a counter-cyclical approach as it

maximizes expected profits. If regulators prevent banks from doing so, banks

will most likely use a pro-cyclical rating approach. This results in excessive

lending in booms at the expense of sharp contractions as soon as defaults start

to rise in recessions. Hence, pro-cyclical estimates of default probabilities result

in pro-cyclicality of lending under risk-based capital requirements. Catarineu-

Rabell/Jackson/Tsomocos conclude that TTC ratings are the first-best solution in

terms of maximizing household and corporate welfare and minimizing default.

Pederzoli/Toricelli/Tsomocos (2010) find by numerical instances of a general

equilibrium model that firms and the household are better-off in terms of expected

utility if PIT estimates are used and if the recession probability exceeds one half.

More specifically, using PIT ratings leads to a stronger reduction in corporate

loan rates and a weaker reduction in deposit interest rates compared to a TTC

rating system. If an expansion is more likely, the opposite is true. Hence, also

in this model PIT estimates affect the economy in a pro-cyclical way. Whether

the bank prefers PIT or TTC estimates when a recession is expected depends


on its role on the interbank loan market. The net lender prefers TTC while the

net borrower PIT. Their results do not necessarily contradict those of Catarineu-

Rabell/Jackson/Tsomocos (2005) as the study of Pederzoli/Toricelli/Tsomocos

(2010) builds upon a richer environment concerning the banking sector.

To summarize, there is empirical as well as theoretical evidence that the estimation

technique used to determine the risk weights under Basel II influences the cyclicality

in required capital and in the lending volume. In particular, PIT ratings should be

deemed to add to the pro-cyclicality of lending compared to the use of TTC rating

systems. But it is not the aim of this thesis to examine the impact of loan rating

systems on total lending and other decisions. Rather, the effects of capital-adequacy

rules as such compared to a regime of laissez-faire are of main interest.

1.3.5 Further Theoretical Research on the Pro-Cyclicality

of Basel II

Buhler/Koziol (2005) address the question of pro-cyclicality in lending by

considering an economy with three agents: a firm, an investor (household), and

a bank. All these agents have constant absolute risk-aversion and have unlimited

liability. The firm’s return on its real investment, the return on the loan, and on the

deposit redemption are exogenously given and normally distributed. Specifically,

the returns of the bank’s asset (“loan”) and liability side (“deposit”) are linked by

an exogenous correlation. The bank is regulated by a VaR approach. They detect

counter-cyclical effects of regulation if there are shocks which affect the firm’s return

risk or the firm’s productivity rate. Pro-cyclical effects occur if the firm’s equity

changes. Hence, whether regulation amplifies shocks in a pro-cyclical manner or not

depends on the sort of shock that has affected the borrower’s risk profile.

Also Zhu’s (2008) study that is based on a dynamic bank model questions the notion

that capital requirements have an inherent tendency to enforce the cyclicality in

lending. The calibrated model shows that banks that are not effectively constrained

by capital requirements exhibit stronger cyclical swings in lending during upswings

than those banks that are constrained by regulation. Constrained banks, however,

hold a capital buffer beyond what is required so that some of the cyclicality may be

absorbed by total capital.15

Repullo/Suarez (2008) analyze the effects of capital requirements within a two-

period framework. The bank is risk-neutral and managed in the interest of the

15Cf. the discussion under Section 1.3.3.

1.4. FURTHER REGULATIONS AND PRO-CYCLICALITY 23

shareholders. Deposits are treated as fully insured debt. They draw comparisons

between the Basel I, the Basel II, and a laissez-faire regime. Loan interest rates,

capital ratios, and capital buffers are the most volatile under the Basel II framework.

Their volatilities increase with the volatility of the exogenously given credit risk. A

shortcoming of the model is that the absolute magnitude of the loan volume is

exogenously fixed.

Tanaka (2002) incorporates credit risk in a stylized way into an IS/LM framework

and shows that after an increase in the overall default probability aggregate output

contracts stronger under Basel II than under Basel I. However, if default probabilities

are high, the loan supply is less sensitive to changes in the loan interest rate under

Basel II than it is the case under Basel I.

Danıelsson/Shin/Zigrand (2004) analyze the persistence of shocks within a simulated

multi-period, sequence equilibrium model. Under VaR-based capital requirements,

shocks are more persistent than in the laissez-faire equilibrium. Moreover, regulation

exacerbates price fluctuations.

1.4 Further Regulations and Pro-Cyclicality

This thesis deals with the pro-cyclical impact of capital requirements. However,

there are also some other rules and regulations that are supposed to amplify credit

cycles.

Accounting and loan-loss provisioning rules have been blamed for enhancing pro-

cyclicality in one way or another.

Bouvatier/Lepetit (2008) find that non-discretionary loan-loss provisioning affects

lending in a pro-cyclical manner. As explanation, they claim banks to behave

myopically. That is, they make little provisions in upswings and face increasing

provisions due to increasing losses in downswings. Furthermore, there is little

evidence of counter-cyclical provisioning behavior which could mitigate the problem.

In particular, poorly capitalized banks do not tend to build provisions in good times.

In this vein, the study of Bikker/Metzemakers (2005) shows that GDP growth and

provisioning are negatively related to each other. Small banks as well as large banks

are found to base provisioning on past losses which is considered as pro-cyclical.

Also Lindquist (2004) confirms that there is little propensity of banks to increase

provisions in good times. Lindquist (2004) gives evidence that unspecified loan loss

provisions and capital buffers seem to be substitutes. However, Bikker/Metzemakers


(2005) and Handorf/Zhu (2006) can also confirm patterns of income-smoothing via

provisioning for some banks.

Another issue is fair-value accounting which has been held responsible for

aggravating the downturn in fixed-income markets. Price drops in some markets

have been accused of melting down banks’ capital positions as the values of held-

for-sale securities had steadily declined. Even before the recent financial crisis that

has been started in 2007, Taylor/Goodhart (2004) pointed to the issue of pro-

cyclicality through the introduction of fair-value accounting standards as laid down

in IAS 39. Issues, that may lead to a “double squeeze” (Taylor/Goodhart 2004,

p. 16) entail the accounting of regulatory capital and the determination of fair

values of non-marketable assets, i.e. in particular loans. Enria et al. (2004) assert

in their simulation study that losses could be doubled in adverse stress scenarios

under full fair-value accounting compared to the accounting principles having been

in place then (cf. Enria et al., 2004, Table 5, p. 23).

In a nutshell, the current practice of provisioning does not seem to be able to balance

credit cycles via equity smoothing whereas there are indications for increasing

cyclicality through fair-value-accounting.

1.5 Risk-Taking, Systemic Risk, and Capital

Requirements

1.5.1 Basel I and Risk-Taking

Beyond increased pro-cyclicality in terms of higher fluctuations in bank capital,

lending, and prices or interest rates, respectively, there have been concerns that

capital requirement do not curtail, as intended, but increase risk-taking, both on

the individual as well as on the sector-wide level. Regulators are concerned with

risk-taking as banking regulation is actually in place to reduce the risks the financial

sector takes. Enforcing financial stability and depositors protections are actually the

main rationales for regulating banks.16

In particular, flat capital requirements, as given by the Basel I Accord, have been

often blamed for setting wrong incentive such that banks take as much risk as

possible given a fixed amount of regulatory capital. The increased risk sensitivity

of the Basel II Accord was intended to remove these deficiencies. Yet, the mistrust

16Cf. Dewatripont/Tirole (1994, p. 31f), Freixas/Rochet (1997, p. 264), and Santos (2000).

1.5. RISK-TAKING, SYSTEMIC RISK, AND CAPITAL REQUIREMENTS 25

in Basel II concerning the issue of increased risk-taking has not disappeared.

A required minimum of capital establishes a minimum risk-participation of equity

holders and elevates the bank’s default barrier. Thus, a required minimum of capital

serves as a rationale why capital regulation is in place. In particular, this was why

the introduction of the Basel I capital requirements were welcome after decades

of eroding bank capital ratios. Basel I was supposed to curtail risk-taking and to

strengthen capital ratios according to Jacques/Nigro (1997). In particular, many

US-American banks had faced large losses in asset values and thus declining capital

ratios at that time after the savings and loans crisis in the 1980ies.

Calling for a minimum amount of capital is considered to be an important

regulatory goal since the bank equity holders’ limited liability results in risk-loving

behavior (Sinn, 1982; Gollier/Koehl/Rochet, 1997; Goodhart, 1996, p. 12f). As

a consequence, low-capitalized banks take excessive risks, known as “gambling for

resurrection” (Rochet, 1992, Calem/Rob, 1998, and Blum, 1999). In light of the

bank failures in 2008, a maximum-leverage rule, irrespective of risk, is gaining

popularity among G-20 leaders (G-20 Leaders’ Statement, 2009).

1.5.1.1 Empirical Evidence and Criticism

Shrieves/Dahl (1992), Jacques/Nigro (1997), Aggarwal/Jacques (1998), Ediz/Mi-

chael/Perraudin (1998), and Rime (2001) are among the scholars who confirmed a

positive relation between capital and the risks assumed.

More specifically, Jacques/Nigro (1997), Rime (2001), and Altunbas et al. (2007)

find evidence that capital requirements effectively prevented banks from increased

risk-taking. Ediz/Michael/Perraudin (1998, p. 20) and Rime (2001) claim that

capital requirements induced banks to raise their capital ratios, irrespective of

internal capital targets. In contrast to the other articles cited above, Shrieves/Dahl

(1992) analyze pre-Basel I bank data. They report a strong and positive relation

between capital and risk-taking among US banks in the years 1984 to 1986 and

conclude that market discipline works well. Following this Shrieves/Dahl (1992),

Heid/Porath/Stolz (2004) find evidence that German savings banks try to keep a

given level of buffer capital in accordance with their risks. They consider the period

from 1993 to 2000.

These views on the relation between risk-taking and the amount of capital held

have been criticized for being too static and for disregarding new techniques in

credit finance.


Jackson et al. (1999) already report exponentially increasing volumes of

securitization in the USA and Europe, recognized as a method to reduce required

capital. Over the years, this way of economizing on capital for given levels of risk,

that has become known as “regulatory capital arbitrage”, has attracted increasing

attention and caused concerns (Jones, 2000; Franke/Krahnen, 2005). Nowadays,

securitization has been made responsible for decreasing lending standards and thus

for having laid the grounds amongst others that finally cumulated to the sub-

prime crisis (Dell’Ariccia/Igan/Laeven, 2008). Moreover, the contagious effects of

securitization have been revealed by this crisis (Gorton, 2008).

1.5.1.2 Theoretical Criticism

The possibility that capital requirements can imply the choice of rather risky

portfolios has brought about a large strand of literature on its own.

One topic has been the interplay between capital requirements and deposit insurance

and their complementary effects.

Bond/Crocker (1993) show that optimal insurance premia depend on the bank’s

capitalization, that serves as an indicator of its probability of default, and that

the optimal insurance plan does not fully cover losses in the bankruptcy state.

The latter implies that there is a role for monitoring bank capital by depositors,

preventing moral hazard. Similarly, Buser/Chen/Kane (1981) prove that banks can

raise their value under flat deposit insurance premia by increasing their leverages

without bounds.17 Further research on the optimal mix of deposit insurance premia

and capital requirements such that the implicit subsidies given by tax payers are

minimized has been performed by Freixas/Gabillon (1999).

These findings serve as a rationale to explain why the deposit insurer seeks to

tax insured institutions implicitly, be it by capital requirements or by threatening

institutions to withdraw charters.

Despite of this kind of complementarity between deposit insurance and capital

requirements there have been concerns that there are still bad incentives for banks,

in particular regardless of the amount of equity capital required, such that banks

hold riskier portfolios under regulation than they would do without. In this vein,

Koehn/Santomero (1980) demonstrate that a stricter, but flat capital-to-asset ratio

17Amongst others, Bhattacharya/Boot/Thakor (1998, pp. 755-757) and Santos (2000, p. 16)review the implications of flat deposit insurance premia on the banks’ risk-taking behavior as wellas the role of risk-based capital requirements to risk-based deposit insurance.


has ambiguous effects on the bank’s probability of default if deposit insurance premia

are constant. Similar conclusions are drawn by Gennotte/Pyle (1991).

But these negative effects of fixed rate deposit insurance contracts can be mitigated

by introducing risk-sensitive capital requirements because they impose an upper

bound on the probability of bank solvency, as Kim/Santomero (1988) show.

Furlong/Keeley (1989) show that also flat capital requirements reduce the banks’

risk-taking if the option value of deposit insurance, that implies a subsidy to the

bank, is properly accounted for. More precisely, they show that the marginal value

of the deposit insurance option declines with increasing asset risk if the capital ratio

rises.18

But flat deposit insurance need not to be the sole source of increased risk-taking.

Even if the bank’s liabilities bear risks that are correlated with the bank assets’

returns, neither a constraint on leverage nor on portfolio composition alone constrain

the bank’s probability of ruin, as Kahane (1977) shows within a mean-variance

framework.

Rochet (1992) shows that choosing risk weights such that they are proportional to

systematic risks helps to decrease the bank’s probability of failure, if banks behave

as portfolio managers and maximize expected utility. Limited liability requires

an additional minimum level of capital in order that the bank is prevented from

choosing inefficient portfolios. He criticizes earlier work on risk-taking for two

reasons: for the assumption of complete markets being used to price the option

value (cf. Kareken/Wallace, 1978; Dothan/Williams, 1980), which is incompatible

with the existence of banks, or for neglecting limited liability as done by the portfolio

models.

Calem/Rob (1999) consider the problem of risk-taking within an infinite-horizon

framework. They consider a deposit insurance scheme that charges the bank an extra

fee if its capital falls short the regulatory thresholds and that charges a flat premium

else. If this deposit insurance scheme is in place, there is a U-shaped relation between

capital and risk-taking. Low-capital as well as high-capital banks take excessive

risks. The premium surcharges or tighter capital requirements aggravate the problem

of risk-taking. Risk-sensitive capital requirements19 are not sufficient to reduce

18Furlong/Keeley (1989) and Keeley/Furlong (1990) criticize in this respect the mean-varianceframework as used by Koehn/Santomero (1980) and Kim/Santomero (1988) amongst others. Inparticular, they criticize that thus the option value implied by deposit insurance has been neglectedso far. Merton (1977, 1978) was the first who characterized deposit insurance costs by a put optionon common stock.

19Risk-based capital requirements linearly increase in the amount invested into the risky assetsas soon as a fixed proportion of risky investments is surpassed. This rule is meant to mimic the


risk in their framework, but uninsured deposits (with depositors being risk-neutral)

discipline the bank, i.e. reduce risk-taking for low capital levels compared to the

case of insured deposits.

Blum (1999) was the first to examine inter-temporal effects analytically within a

two-period framework. Assuming deposits are insured at a fixed-rate, a bank takes

more risk if the capital requirement is binding in the first period since bank capital

thus becomes more worthy in the second period. Contrary to Calem/Rob (1999) and

others, Blum (2002, p. 1429) challenges the view that subordinated debt prevents

banks from excessive risk-taking. Rather, he claims, the higher the contracted

interest rate on that debt, the more valuable the ‘option to go bankrupt’ becomes,

and the bank takes even more risk if it does not commit to a level of risk ex ante.

1.5.2 Basel II and Risk-Taking

1.5.2.1 The Three Pillars of Basel II

As a consequence of these rather disappointing results, the academic literate

has begun to pay more attention to risk-sensitive capital requirements. But as

many studies suggest that even risk-sensitive capital requirements alone do not

prevent banks from increased risk-taking, the other two pillars of Basel II besides

capital requirements, market discipline and auditing (supervision), have gained in

importance. Uninsured bank debt is considered as to foster market discipline.

In particular, Calem/Rob (1999) and Decamps/Rochet/Roger (2004) call for

uninsured debt whereas Milne/Whalley (2001) call for continuous auditing to

prevent low-capitalized banks from gambling for resurrection. Also Dangl/Lehar

(2004) show within their continuous-time bank model that VaR-based capital

requirements reduce the auditing frequency such that the IRB approach should

be regarded as superior to the Standardized Approach or the Basel I Accord.

Benink/Wihlborg (2002) ask for mandatory subordinated debt to strengthen market

discipline and disclosure of risks. Blum (2002) questions this proposal. He argues

that once the conditions for issuing subordinated debt are contracted, the bank

with limited liability considers the associated costs as sunk and may even take more

excessive risk. Only ex ante credible commitments can go against this problem.

In contrast, Jokivuolle/Vesala (2007) still find positive effects of Basel II without

considering the other pillars. They show that risk-based capital requirements

Basel-I requirements.


alleviate the problem of risk-taking and cross-subsidization of loan rates in a

competitive loan market where asymmetric information between lenders and

borrowers is present.

1.5.2.2 Systemic Risk

Aside from risk-taking on the single-bank level, the impact on risk-taking within

the whole banking system has attracted increased attention. This research question

has been motivated by the idea that an altered interdependence between banks

could arise because of capital requirements such that more banks are prone to other

banks’ insolvencies or to adverse market outcomes while their own risk appears to be

reduced. Thus, the inherent fragility of banking (Diamond/Dybvig, 1983) and bank

net works (Allen/Gale, 2000) in the absence of regulation could not be addressed

by capital adequacy rules.20 Even worse, regulation would jeopardize its aim of

enhancing financial stability.

Eichberger/Summer (2005) study the trade-off between the risk exposure of single

banks versus the risk-exposure on the interbank market and how it is affected by

capital requirements. Although capital requirements unambiguously reduce risks

on the single-bank level in their model (there are no adverse portfolio allocations

possible as banks can grant loans to one single firm only), risks from loans

granted in the interbank market may increase for the following two reasons: first,

banks operating under a binding capital requirement will increase lending on the

interbank market since it is considered as riskless by both banks and regulators ex

ante. Consequently, interbank loan rates are lower under regulation than without

regulation. Therefore, initially well-capitalized banks, for which regulation is not

binding, increase lending to their local firms by increased borrowing on the interbank

market since there are no other borrowing opportunities for banks. Equity and

deposits are exogenously fixed. Thus, shocks in the real sector are more likely to

damage remote banks under regulation than without regulation.

In contrast, there are also arguments especially for risk-sensitive capital requirements

to reduce systemic risk. Acharya (2001) shows that banks with limited liability tend

to take correlated investments, thus increasing systemic risk. To counterbalance

this effect, capital requirements should account “sufficiently” for correlations (ibid.,

Proposition 7, p. 29). Consequently, a fixed-weight regime for capital requirements

20Deposit insurance which is nowadays in place in many countries (cf. BCBS, 1998) probablyprevented banks from runs by retail depositors by and large during the recent crisis. However, thebanking system has been subject to a run by wholesale investors, as Gorton (2009) points out.


cannot mitigate this behavior and hence cannot reduce systemic risk. Yet, it is still

to be proven if the IRB-formulæ sufficiently account for real-world correlations.

Danıelsson/Zigrand (2008) call for capital requirements, too, since in general

equilibrium excessive risk-taking occurs at the expense of other market participants’

risk positions. At the heart of this mechanism lies the notion that everyone considers

its own impact as negligible. VaR-based capital requirements can mitigate systemic

risk, but the possibility that markets cannot clear increases, as a sufficient number

of agents may lose their ability to absorb risks, i.e. to trade. Moreover, liquidity

decreases, asset price volatility may rise, and price co-movements may appear even

if asset prices are stochastically uncorrelated.

Chapter 2

Framework of the Analysis

2.1 Exogenous Shocks and the Business Cycle

A bank or an investor in general may be affected by different types of shocks. As

pointed out by Allen/Saunders (2003, p. 2) any profit or loss observed can be caused

by an ex ante shift in the return distribution or it is simply due to a realization based

on a fixed loss distribution. We will refer to the latter as realized shocks and to the

former as expectation shocks. We will treat all kind of shocks as exogenous. This

classification will be useful for characterizing the results obtained in this thesis. Also

the results obtained by Buhler/Koziol (2005) can be distinguished according to this

classification of the underlying shock. They present how realized losses, i.e. decreases

in capital, imply pro-cyclical effects whereas shocks affecting risk and productivity

imply counter-cyclical reactions.

2.1.1 Equity Shocks

In the one-period models, equity shocks are understood as exogenous changes in

the bank’s initial equity: initial equity is considered to be lower than in the base

case if a negative shock has occurred the period before, which has resulted in losses

that had to be borne by the bank, hence reducing equity. Conversely, initial equity

that is higher than in the base case, is associated with profits made in a notional

previous period. Beyond that, different levels of bank equity may be linked to the

state of the business cycle as pars pro toto: high levels of equity represent a boom

state whereas low levels of equity reflect a bust. By this sort of co-movement of

bank equity with the state of the business cycle, changes in bank equity will be

considered as pro-cyclical. Consequently, it makes sense to refer to co-movements

31

32 CHAPTER 2. FRAMEWORK OF THE ANALYSIS

of other magnitudes with bank equity as pro-cyclical too. A formal definition will

be given below. The level of bank equity is thus a backward-looking variable that

characterizes the state of the cycle and the soundness of the bank.

2.1.2 Expectation Shocks

Expectation shocks comprise changes of default probabilities, expected returns

on projects, project volatilities, and of correlations. Productivity shifts will also

be considered as expectation shocks. All these variables contain the agents’

expectations and knowledge about the ability of single firms to honor their debt

obligations. They form the credit worthiness of the whole economy and thus

affect the riskiness of loans and deposits. Shifts in these variables reflect changes

in the agents’ minds about future prospects. Lower expected returns and lower

productivity are associated with an economic downturn. The reverse holds true

with respect to volatilities and correlations as increasing values of volatilities and

correlations are linked with rising uncertainty about future outcomes. Moreover,

higher correlations hamper diversification. To summarize, expectation shocks refer

to the future state of the cycle and affect the economy’s fundamentals.

2.1.3 Interference of Different Shocks

Of course, in reality, we may be confronted with several different shocks at once. But

it is unclear to what extent a given negative shock is due to shifts in distributions,

or to what extent it is just bad luck. Economic subjects may update their beliefs

after a negative or positive shock has realized. A behavior such as this complicates

things further, as expectation shocks may follow realized shocks (i.e. equity shocks)

whatever the reasons for the latter might have been.

However, considering different shocks at once does not help understand the impacts

that regulation might have on the economy since different mechanisms will come

into play at once. Furthermore, it is ex ante not clear, how different types of shocks

should be mixed in a meaningful way. In reality, even countervailing shocks may

come into force simultaneously. Yet, shocks may also arise independently.

For these reasons, we will confine ourselves to analyze regulatory effects for isolated

shocks within the one-period frameworks considered.

Within a two-period model, the interference of expectation and realized shocks

occurs as follows: in the first period, the economy faces a shock concerning expected

2.2. DIFFERENT TYPES OF CYCLICAL BEHAVIOR 33

magnitudes. Consequently, the distribution of the bank’s equity in the middle of

the time elapsed is affected and so is the distribution of final outcomes. Hence,

expectation shocks in t = 0 imply on average realized shocks in t = 1.

2.2 Different Types of Cyclical Behavior

Having started with different types of shocks, we would like to classify the reactions

of endogenous variables toward these exogenous shocks. This classification shall

serve to distinguish some of the various impacts regulation may have on the behavior

of endogenous variables. Endogenous variables may encompass loan and deposit

volumes, deposit interest rates and portfolio effects. Classifying those impacts also

helps to clarify our notion of pro-cyclicality of regulation, cf. Buhler/Koziol/Sygusch

(2008), compared to other concepts that can be found in the literature. Above all,

we do not ask if (regulatory) capital is pro-cyclically affected by regulation or not,

such as some authors do (cf. Estrella, 2004; Goodhart/Hofmann/Segoviano, 2004;

Gordy/Howells, 2006, and Kashyap/Stein, 2004). Rather, we will ask how changes in

the bank’s initial equity will affect endogenous variables with and without regulation.

That is, changes in the bank’s equity are treated as exogenous shocks amongst others

whose effects ought to be analyzed.

Let θ be an exogenous parameter whose change represents either a realized or an

expectation shock. Consider the parameter values θ1 and θ2, θ1 < θ2, belonging

to any given interval [θ, θ], θ < θ, on which regulation is binding. Let Θ be an

endogenous variable where Θ∗1 denotes the optimal equilibrium outcome without

regulation at parameter value θ1 and Θr1 that under a given regulatory regime at θ1.

Likewise, we define Θ∗2 as well as Θr2. Consider the differences

∆θ = θ2 − θ1 ,

∆Θ∗ = Θ∗2 −Θ∗2 ,

∆Θr = Θr2 −Θr

1 .

Regulation may refer to any concept of capital requirement.

Then we call regulation pro-cyclical on [θ, θ] if

∆Θr

∆θ>

∆Θ∗

∆θ≥ 0 or −∆Θr

∆θ> −∆Θ∗

∆θ≥ 0 (2.2.1)

holds for all θ1, θ2 ∈ [θ, θ], θ1 < θ2.


This definition captures all those situations where endogenous variables under

regulation are affected more strongly by a shock than those without regulation.

Moreover, this notion of pro-cyclicality is always linked to amplifications that go

into the same direction under both regimes. Thus, by pro-cyclicality we understand

either contractions or expansions that are enforced by regulation.

We say that regulation is not pro-cyclical on [θ, θ] if

∣∣∣∣∆Θr

∆θ

∣∣∣∣ ≤∣∣∣∣∆Θ∗

∆θ

∣∣∣∣ and

[sgn

(∆Θr

∆θ

)= sgn

(∆Θ∗

∆θ

)or

∆Θr

∆θ= 0

](2.2.2)


This definition spans those situations where the endogenous variable with and

without regulation is affected in the same direction but the effect for the case without

regulation is stronger. Thus, if regulation is not pro-cyclical, regulation dampens

endogenous cyclical effects.

Regulation is said to be counter-cyclical on [θ, θ] if

∆Θr

∆θ< 0 <

∆Θ∗

∆θor

∆Θr

∆θ> 0 >

∆Θ∗

∆θ(2.2.3)


Counter-cyclicality includes all cases where the endogenous variable under regulation

reacts in the opposite direction compared to the case where the bank is unregulated.

Finally we call regulation on average pro-cyclical on [θ, θ] if

sgn

(∆Θr

∆θ

), sgn

(∆Θ∗

∆θ

)∈ 0, 1 ∀ θ1, θ2 ∈ [θ, θ] and

∆Θr

∆θ>

∆Θ∗

∆θ≥ 0

or if

sgn

(∆Θr

∆θ

), sgn

(∆Θ∗

∆θ

)∈ −1, 0 ∀ θ1, θ2 ∈ [θ, θ] and

−∆Θr

∆θ> −∆Θ∗

∆θ≥ 0 (2.2.4)

holds for all θ1 ∈ [θ, θ) and for θ2 = θ.

This definition is weaker than Definition (2.2.1) because pro-cyclicality on average

may allow for subsets of [θ, θ] on which the unregulated endogenous variable reacts

2.3. THE MODEL’S STRUCTURE 35

more strongly on a given shock than the regulated one does. The latter may emerge

by differences in convexity or concavity, respectively. 1

2.3 The Model’s Structure

2.3.1 The Basic Set-Up

We aim at assessing the consequences that specific shocks as classified above have on

the amount of lending when minimum capital requirements are binding compared to

the case of a laissez-faire economy. Contrary to the majority of the literature, we do

not only consider our analysis on the relation between the borrowers and the bank,

but track the financial linkages further to the bank’s creditors, the households.2

Therefore, we consider a three-sector economy comprising of firms, a bank, and a

household. The firms operate on a perfectly competitive goods market. The firms’

risks are related to output and/or price fluctuations as the demand of their specific

product may change. These mechanisms are not modeled explicitly. Particularly,

the model is not closed by equating the firms’ supply of goods with the household’s

demand. Instead, our framework concentrates on the financial linkages to the banks

and finally to the household.

The household can invest its initial endowments into a riskless asset yielding an

exogenous, riskless interest rate and into bank deposits. Deposits are not insured.

Thus, the credit risk of the bank’s loan portfolio translates into the pay-offs of

the deposits. Early withdrawal of deposits is excluded. Therefore, the deposits

considered are in fact a standard one-period debt contract on which the bank may

default in some states of the world. The household is risk-averse and maximizes its

utility over final wealth.

The bank’s exclusive access to firms and its exclusive role as a financial intermediary

between firms and the household is given by assumption. The bank has monopoly

power on the loan market and the deposit market. The bank can only grant

corporate loans and issue risky deposits. Thus, this model only allows an analysis

of shifts between corporate loans, but not between corporate loans and risk-free

1Results 15 and 26 will present conditions where regulation has on average pro-cyclical effects.Examples in Section 6.6, illustrated Figures 6.9 and 6.10 provide evidence of the reasonability ofthis definition.

2Of course, the literature that is concerned with the fragility of banking and depositconvertibility concentrates on the relationship between the bank and its depositors. However,the focus is there on liquidity problems and other risks, such as credit risk, are neglected.


sovereign bonds as it was perceived in the 1990/91 Credit Crunch in the US

(Haubrich/Wachtel, 1993, p. 3f; Berger/Udell 1994, p. 586; Furfine, 2001, p. 34).

The firms apply for credit in a situation of perfect competition amongst each other

such that their expected profits are driven to zero in equilibrium.

The risk-neutral bank maximizes expected final wealth by simultaneously choosing

the loan interest rates, the composition of the loan portfolio and the deposit interest

rate. The deposit interest rate does not only depend on the deposit volume but also

on the composition of the loan portfolio since the latter is a key determinant for the

default risk of the deposit. We assume that the bank can reliably commit to every

arbitrary loan portfolio composition which ultimately shapes the state-dependent

pay-offs anticipated by the household. There is no asymmetric information between

firms, banks, and households.

Loan and deposit contracts are fixed at the beginning of the period and are paid off

at the end of the period when uncertainty is resolved. Both the firms and the bank

may default on their obligations and have limited liability. If the loan redemptions

do not cover the promised repayment amount of the deposits, the bank defaults and

the depositor obtains the total loan redemptions. Also the risk-free sovereign bond

lasts for one period each. There are no other contracts available.

Below, the assumptions of the bank’s risk-neutral behavior, the restrictions on bank

liabilities, in particular equity finance, and the bank’s market power are further

discussed.

2.3.2 Fixed Bank’s Equity Capital and Uninsured Deposits

We assume the following financial restrictions: bank capital is exogenously fixed

in the one-period model or at the beginning of the first period in the two-period

model whereas the volume of deposits (bank debt) is endogenously determined.

Bank capital endogenously results from past gains or losses only in the two-period

version at the start of the second period. Since deposits are uninsured and since the

household is risk-averse, the bank faces an endogenous trade-off between deposit

volume and deposit costs where the latter depend on the risk of the bank’s loan

portfolio, i.e. the loan portfolio composition. Beyond the bank’s loan policy, deposits

may also become more costly for the bank and deposit supply may decrease after

expectation shocks.

The first assumption reflects that it is hardly possible for banks to raise new capital

in economic downturns due to negative signalling effects or in very short time. This


concern is also shared by the BCBS (BCBS, 2004, Art. 757, c). Instead, banks

will have to adjust their loan portfolios concerning volume and risk which stands

in contrast to the microeconomic approach of Estrella (2004) and the simulation

studies of Goodhart/Hofmann/Segoviano (2004), Kashyap/Stein (2004), and Gor-

dy/Howells (2006). Repullo/Suarez (2008) take an intermediate view in their model.

There, banks can only raise new equity capital if they enter the market or grant new

loans.

In our model framework, deposits are uninsured such that there are feedback effects

from the risk-averse depositor to the bank if overall risk increases or the bank

chooses specifically risky portfolios so as to counterbalance a constrained business

volume which may be the case under binding capital requirements. This feature

is novel to the literature on banks which can be distinguished concerning credit

risk of deposits/bank debt or the role of depositors along the following lines: first,

if the bank issues uninsured debt, the counter-party is risk-neutral with infinite

elasticity of the deposit supply (Dermine, 1986, p. 107; Calem/Rob, 1999, p. 320 and

pp. 342-346; Zhu, 2008, p. 173). Second, independent of the final risk-characteristics

of the pay-offs from the deposit contract, portfolio choice models treat the risk

transmission from the asset to the liability side as exogenous, by assuming assets

and liabilities are correlated by an exogenous parameter (Hart/Jaffee, 1974; Kahane,

1977; Kim/Santomero, 1988, and Buhler/Koziol, 2005). This modeling technique

can be also found with continuous-time models, as in Pennacchi (2005). Third,

some papers assume exogenously given deposit cost functions (e.g. Blum, 1999) or

exogenously given deposit-supply functions (Klein, 1971, Monti, 1972, and Dermine,

1986, p. 102). Lastly, the vast majority of papers which treat deposits as insured do

not model the deposit supply side at all (Estrella, 2004; Eichberger/Summer, 2005;

and all simulation-based studies cited so far).

Since there are no informational asymmetries between the bank and the household,

the bank credibly commits to its loan allocation and hence to the level of risk

which in turn determines the deposit interest rate and the feasible deposit volume.

So the bank must incur the costs of its level of risk-taking and cannot shift these

costs to other agents as a deposit insurance corporation such that there are market-

disciplining effects.

Furthermore, bank equity will play the role of a buffer in case of bankruptcy. Since

the household is fully informed, it can perfectly assess this role of bank equity in

case of the bank’s failure with respect to residual pay-offs. Bank equity thus serves

as a device to share risks between the bank (or its managing owners) and the debtor


(here the household). This function has been formerly analyzed by Gertler/Hubbard

(1993) for firms in general that issue uninsured debt.

2.3.3 The Bank’s Risk-Neutrality

The interaction between the household and the bank implies that the risk-neutral

bank does not pick the loan with the highest expected return but diversifies credit

risk in a non-trivial way.

In contrast, diversification of risk has often been achieved in past models by

introducing risk-aversion at the bank level so that the bank maximizes expected

utility. An important strand of this kind of literature are the so-called portfolio

models. This approach has been pursued by Hart/Jaffee (1974), Kahane (1977),

Koehn/Santomero (1980), Kim/Santomero (1988), and Buhler/Koziol (2005),

amongst others.

Santomero (1984, p. 582) states that models that work under the assumption of

risk-aversion implicitly build on the notion that the bank is run by managers

who cannot diversify their human capital or that the bank is owned by investors

who have limited access to other investment opportunities. Conversely, he claims

that the assumption of risk-neutrality can be aligned with banks being owned

by investors who have broad access to further investment opportunities such that

diversification is no longer aimed at the bank’s portfolio level, but at the investors’

asset universe. However, the assumption of risk-neutrality has also been simply

viewed as an approximation (Baltensperger, 1980, p. 25). So far, many models

with a risk-neutral banking firm have evolved, such as Blum (1999), Calem/Rob

(1999), Dermine (1986), Eichberger/Summer (2005), Estrella (2004) Klein (1971),

Monti (1972), Repullo/Suarez (2008), Suarez (1994), Thakor (1996), and Zhu

(2008), to name a few. In contrast, risk-averse banks and other risk-averse

financial agents can also be found in general equilibrium approaches. In Catarineu-

Rabell/Jackson/Tsomocos (2005), firms and banks have µ-σ preference functions as

objective function. Similarly, banks in Danıelsson/Zigrand (2008) and traders in

Danıelsson/Shin/Zigrand (2004) maximize expected utility.

2.3.4 Market Power in Banking

In our model, the bank has monopoly power on both the loan and deposit market,

thus standing in this respect in the tradition of the neo-classical models of Klein


(1971), Monti (1972), Dothan/Williams (1980), Dermine (1986), and Blum (1999)

who all consider a double-sided monopoly as well.3 But by the endogenous risk-

transfer and the endogenous interaction between a risk-neutral bank and a risk-

averse depositor, we considerably depart from their frameworks. Let us review some

reasons and evidence of market power in banking.

The bank’s monopoly power on the market for loans can be explained as a result of

relationship banking: once firms are bound to a given bank, the bank can extract

the monopoly rent on loans (Sharpe, 1990, p. 1069; Repullo/Suarez, 2008).

Though relationship banking is deemed a convincing argument for monopoly power

on the loan market, monopoly power on the deposit market is seen as rather

controversial as depositors are considered to have access to a sufficiently high number

of alternative banks. If deposits are fully insured, vagueness about the bank’s asset

quality should be negligible even in a world of asymmetric information and, at

least concerning term deposits, perfect competition should emerge. A counter-

argument frequently put forward is about locally fragmented markets such that

alternatives to depositors are too scarce for sustaining a competitive environment

(e.g. Saunders/Schumacher, 2000). In this vein, there are also authors who assume

monopoly power of banks exclusively on the deposit market (Kareken/Wallace, 1978;

Rochet, 1992, p. 1143; and Suarez, 1994). Recent technological advances, such as

internet-based banking, may weaken this argument, but may not render it obsolete

as long as there is demand for individual consultations.

Greenbaum/Thakor (2007, p. 108) explain market power of banks on the deposit

side by two sources that are related to the bank’s size. First, economies of scale arise

by diversification: the bigger the bank, the more loans can be made and the more

granular the loan portfolio becomes. Thus risks can be reduced with increasing size.

Risk reduction, in turn, makes the depositors better-off, even if they are risk-neutral

as they face a concave pay-off from uninsured debt. As a result, the offered deposit

interest rate is the lower, the bigger the bank is. With increasing size, banks may

reduce deposit interest rates stronger than the reduction of risk justifies. Single

depositors will accept, as long as they have no similar diversification possibilities.

Second, credit analyses associated with the loans made are done once by the bank

and are borne by many depositors. Otherwise, if there is no bank, the same costly

screening efforts will be performed directly by the depositors (then simply lenders).

As coordination generally fails, these efforts will partly be done more than once,

3As the bank is actually on the demand side of the deposit market, the bank’s behavior should berather termed as monopsonistic (cf. Mas-Colell/Winston/Green, 1995, p. 500; Hendersen/Quandt,1958, p. 164).


raising the costs per deposit.

These economies-of-scale argument work when the number of depositors per bank is

large and if there is no deposit insurance. As result, the bank is a natural monopoly.

Thus, the question whether there are monopolistic structures in banking or not is

an issue of institutional settings, of arrangements in banking regulation, and in the

end of empirical evidences. Baltensperger (1980, p. 18) argues that “probably [...]

specific circumstances” may justify whether or not the assumption of monopolistic

behavior can be justified for modeling banks.

Evidence is given by Keeley (1990) who identifies bank charter values with

market power on loan and deposit markets. However, declining capital ratios in

the USA are seen as if bank charters lose their value or, equivalently, market

power decreases. Similarly, Neven/Roller (1999) confirm a cartel-like conduct of

banks concerning granting mortgages, but at a decreasing rate over time. Both

Keeley (1990) and Neven/Roller (1999) associate the declining market power

with deregulations. Further evidence of anti-competitive behavior is found by

Molyneux/Lloyd-Williams/Thornton (1994) for Germany and de Bandt/Davis

(2000) for small banks in Germany and France. Gischer/Stiele (2009) find that

the locally segmented market of savings banks in Germany is characterized by

monopolistic competition.

To summarize, this model framework allows us to study a bank’s simultaneous choice

of the risk allocation concerning potentially correlated investments and on deposit

costs which are captured by the interest rate on deposits. Furthermore, the size of

the bank, i.e. the single loan volumes and the deposit volume endogenously emerge.4

In particular, credit risk is endogenously transferred to the household, resulting in

a feed-back effect for the bank when it takes deposit supply as given for maximizing

its own expected wealth.

This model framework then serves to compare these decisions under various regimes.

The laissez-faire equilibrium will be compared with the equilibria that occur under

the Standardized Approach and a VaR approach to determine economic capital at a

given confidence level. The latter approach is meant to represent the IRB Approach

as done by Buhler/Koziol (2005), Dangl/Lehar (2004), Danıelsson/Shin/Zigrand

4Contrary to Baltensperger’s (1980, p. 18, 21) criticism on bank models with monopoly powerconcerning deposits, the assumption of monopoly power is not necessary to obtain an endogenousupper bound on the bank’s size in our framework. A competitive market structure between bankand depositor can be modeled by maximizing the household’s expected utility or preference functiongiven a zero-expected-profit condition imposed on the bank. That is, the bank acts on behalf ofthe depositor as otherwise the depositor would choose an alternative bank.


(2004), Estrella (2004), and Danıelsson/Zigrand (2008).


Part II

Bernoulli Distributed Loan

Redemptions

43

Chapter 3

Theoretical Analysis

3.1 Introduction

In this chapter, a specific model that is based on the framework outlined in Chapter

2 is set up and discussed. There are two firms who demand credit, a household that

supplies deposits, and a bank as intermediary between the firms and the household.

The analysis is restricted to one period.

The agents are introduced in the next section, Section 3.2. The firms are presented

in Section 3.2.1. The household’s decision is outlined in Section 3.2.3, the bank’s

in Section 3.2.4. Section 3.2.2 lays the grounds for the household’s and the bank’s

objective function as it highlights the state- and portfolio-dependent pay-offs from

the deposit contract.

Section 3.3 deals with the equilibria with and without regulation. Section 3.3.1

examines the equilibrium without regulation in general and, moreover, highlights

some specific equilibria that can be stated explicitly. Likewise, equilibria under

the Standardized Approach (Section 3.3.2), and under a VaR-based approach

(Section 3.3.3) are discussed. Specific equilibria arising under binding regulation are

compared with specific equilibria arising under the laissez-faire economy presented

in Section 3.3.1. First conclusions on the cyclical impact of regulation are drawn.

Section 3.4 concludes. In particular, Table 3.9 provides an overview over the

results on regulatory impacts through the Standardized Approach, and Table 3.10

summarizes the findings concerning the VaR-based regulatory approach.

The whole Part II is based on Buhler/Koziol/Sygusch (2008), but considerably

extends that study. Results, that can be also found in Buhler/Koziol/Sygusch

(2008), are indicated as such. Part II adds to Buhler/Koziol/Sygusch (2008) by

45

46 CHAPTER 3. THEORETICAL ANALYSIS

the theoretical analysis of the model, i.e. the non-numerical analysis of equilibria

and the thus obtained results on cyclical impacts by regulation. Furthermore,

the numerical analysis performed in Chapter 4 exceeds the study presented by

Buhler/Koziol/Sygusch (2008).

3.2 The Model

3.2.1 The Firms

We assume that there are two firms i, i = 1, 2, where each can run a single risky

project.1 The outcome of each project is Bernoulli-distributed, i.e. a project can

either be successful, Xi = 1, with probability pi, or it can fail, Xi = 0 with

probability 1− pi. Firms exhibit a linear production technology where each unit of

capital used transforms into αi > 1 units of output. Firms may scale their projects

arbitrarily high. Firms are risk-neutral and for simplicity they are exclusively

financed by (bank) debt Li. As a result, their production functions are given as

follows:

αi · Xi · Li, i = 1, 2 .

At the end of the period each loan is repaid conditional on the firm’s success. Due

to the firms’ limited liability, the bank obtains the following payment from each loan

contract:

minLi ·Ri, αi · Xi · Li

, i = 1, 2 . (3.2.1)

where Ri denotes the gross interest rate on the nominal debt volume Li. Ri and

Li are endogenous. By risk-neutrality, both firms invest as long as their expected

profits are non-negative. Therefore, they maximize:

maxLi≥0

E[

maxαi · Xi · Li − Li ·Ri, 0

]

⇔maxLi≥0

pi ·max αi −Ri, 0 · Li .(3.2.2)

Let us assume that both firms accept any loan volume as long as their expected

wealth is equal to or greater than zero and that loan volumes are non-negative.

Loan demand by firm i can then be characterized by the following loan demand

1A short version of this section can be found in Buhler/Koziol/Sygusch (2008, p. 138f).

3.2. THE MODEL 47

correspondence,

Ldi (Ri)

≥ 0 if Ri ≤ αi

= 0 if Ri > αi, i = 1, 2 . (3.2.3)

Consequently, the bank, which supplies loans as a monopolist, does not forgo any

profits if it sets each loan interest rate Ri equal to the marginal productivity αi of

firm i, because the monopolistic bank is still free to scale the loan volume arbitrarily

up and down. Thus, it always chooses2

R∗i = αi, i = 1, 2 . (3.2.4)

The resulting gross interest rate R∗i is independent of the bank’s refinancing

conditions and hence of the deposit market, the household, and the regulation.

We make use of this result from here on without explicitly referring to it.

Firms are not only price-takers on the market for corporate loans, but implicitly

also on the goods market as their productivity is fixed and transforms the financing

conditions linearly into goods. Hence, the firms operate under perfect competition.

Instead of prices equaling marginal cost, the dimensionless marginal productivity αi

equals a dimensionless cost rate, expressed by the gross loan interest rate Ri. Inputs

are capital in form of loans and thus denominated in dollars.

Unless otherwise stated, we will assume firm i = 2 as the riskier firm in terms of a

higher default probability 1 − p2, a higher default variance, and a higher expected

(gross) return p2α2,

(1− p1) < (1− p2), p1(1− p1) < p2(1− p2), p1α1 < p2α2 , (3.2.5)

implying that:

α1 < α2, V(α1X1) < V(α2X2) ⇔ p1(1− p1)α21 < p2(1− p2)α2

2 and p1 >1

2.

Undertaking risky projects shall dominate the risk-free investment yielding the gross

rate Rf :

p1α1 > Rf ≥ 1 . (3.2.6)

Thus, the risk-neutral firms would not invest their capital into the riskless bonds

even if they could.

2Also in Repullo/Suarez (2008), the bank charges the firms the total return in case of successas interest on loans.


Hence, the productivity parameters are bounded from below, αi > 1 by (3.2.6). In

what follows,

αi ≤ 2 ∀i = 1, 2 , (3.2.7)

is additionally assumed as upper bound such that returns from production and hence

on loans, do not become too big. Assumption (3.2.7) helps to directly derive Results

3, 4, 9, 11, and 12. To obtain Result 7, Assumption (3.2.7) is further restricted to

αi ≤ 43, which is only sufficient, however.

Furthermore, the Assumptions (3.2.6) and (3.2.7) result in

p1 > p2 >1

2(3.2.8)

which is not restrictive, as the probability 1−pi represents firm i’s default probability

which is thought to be far lower than a half in this context. Equality, i.e. p1 = p2,

as far as considered, will be deliberately noted.

On the level of individual firms, we have cared so far about how returns of each

project are distributed. In order to decide the allocation of funds among these two

firms, it is necessary to consider the joint distribution of the projects’ returns. As

the marginals are given by pi, i = 1, 2, the joint distribution is restricted to

P(X1 = 1, X2 = 1) + P(X1 = 1, X2 = 0) = p1

P(X1 = 0, X2 = 1) + P(X1 = 0, X2 = 0) = 1− p1

P(X1 = 1, X2 = 1) + P(X1 = 0, X2 = 1) = p2

P(X1 = 1, X2 = 0) + P(X1 = 0, X2 = 0) = 1− p2

.

By introducing the notation

q ≡ P(X1 = 1, X2 = 1

), (3.2.9)

we can rewrite the above system of equations to yield

P(X1 = 1, X2 = 0) = p1 − qP(X1 = 0, X2 = 1) = p2 − qP(X1 = 0, X2 = 0) = 1− p1 − p2 + q

. (3.2.10)

To ensure that the Probabilities (3.2.10) that form the joint distribution are all

positive, the probability q must be bounded by (cf. Joe, 2001, p. 210)

p1 + p2 − 1 < q < min p1, p2 . (3.2.11)

3.2. THE MODEL 49

Note that p1 + p2 − 1 > 0 holds due to (3.2.6) and (3.2.7). Furthermore, Condition

(3.2.11) guarantees that the correlation of both projects, X1 and X2, given by

Corr(X1, X2) =q − p1p2√

p1(1− p1)p2(1− p2), (3.2.12)

is bounded on (−1, 1). A correlation of one between both projects is associated with

p1 = q and p2 = q, whereas both projects are perfectly negatively correlated if and

only if q = 0, p1 = 12, and p2 = 1

2hold. For p1 6= p2, the actual bounds can be quite

apart from −1 and 1. Because of p1 > p2 >12

these bounds are as follows:

−1 < −1− p1

p2

< Corr(X1, X2) <p2

p1

< 1. (3.2.13)

That is, the upper bound is determined by the ratio of the firms’ success probabilities

whereas the lower bound is equal to the ratio of Firm 1’s default probability over

the other firm’s success probability. If success probabilities are rather close-knit

together and close to one, the lower bound is close to zero and the feasible range of

correlations is located asymmetrically around zero: if Firm 1 undertakes its project

successfully with p1 = 0.99 and Firm 2 with p2 = 0.98, their default correlation

ranges at most on (− 198, 98

99).3

3.2.2 The Repayment of Deposits

In this section we analyze the repayments of the uninsured deposits at the end of the

period. As soon as these repayments are fully characterized, the objective functions

of the household and the bank can be analytically presented.

There are two distinct determinants that characterize the default risk of the deposits.

First, and basically, the uninsured deposits bear risks since there are risks on the

bank’s asset side, i.e. by the Bernoulli-distributed loan redemptions discussed in the

last section. Second, the riskiness of the deposits is shaped by the bank’s portfolio

compositions: the relation between the amounts of promised loan redemptions

to each other and their respective relations to the amount of promised deposit

redemption characterize under which states of the world deposits can be redeemed

as promised and under which states the depositor is left with a stake in the residual

3The restrictions on parameter values increase with the number of projects, i.e. marginalBernoulli distributions considered. Chaganty/Joe (2006) set out explicit conditions on parametersfor three- (ibid., pp. 198-200) and four-dimensional Bernoulli distributions (ibid., pp. 202-203). InSection 5.2, we will deal with a multivariate Bernoulli distribution that is set up by a mixturemodel following the general pattern proposed by Joe (2001, pp. 210).


bank portfolio redemption. Deposits are always senior to equity.

Let D denote the deposit volume and RD the respective gross interest rate, i.e. D·RD

denotes the promised repayment of the deposit volume inclusive of the promised

interest. Each promised loan redemption is given by LiRi, i = 1, 2. According to

(3.2.4), the bank’s optimal choice of the loan interest rate is not affected by any

other decisions to be taken. Therefore, we proceed with Ri = R∗i = αi, i = 1, 2.

Because there are no information asymmetries, it is known to the household as the

potential depositor how much equity WB ≥ 0 the bank initially possesses and that

there are only the household’s deposits D ≥ 0 and the bank’s initial equity WB at

the bank’s disposal to grant two loans to two distinct firms. We explicitly allow the

bank to intermediate between depositors and borrowers even if it does not initially

have any equity.4

Thus, each loan volume granted can be directly linked to the bank’s liabilities. So

the balance sheet identity implies

L1 ≡ ` · (D +WB) ,

L2 ≡ (1− `) · (D +WB) ,

L ≡ L1 + L2 ≡ D +WB ,

(3.2.14)

where ` represents the percentage amount of available total capital D +WB that is

devoted as loan to the Firm 1. Henceforth ` will be called the loan-allocation rate.

L is referred to as the total loan volume.

We exclude the possibilities of short-sells, i.e. that the lender-borrower relationships

become reversed. Therefore, the loan-allocation rate is restricted to the unit interval,

` ∈ [0, 1] . (3.2.15)

The probability that deposits are paid back as promised depends crucially on

the relations amongst the promised deposit repayment, DRD, and promised loan

repayments, Liαi, i = 1, 2. Two risky loans and a single debt instrument on

one balance sheet imply that we must consider four cases for analyzing the state-

dependent pay-offs that arise from the deposit contract. Furthermore, we rule out

α1R1+L2α2 < DRD since the bank will then go bankrupt with certainty, resulting in

an expected final wealth of zero for the bank. Hence, this choice is clearly dominated

by any other policy. The objective function of the bank owners is presented in

4It is not until the two-period framework in Chapter 7 that, at beginning of the second period,zero equity is exclusively associated with bankruptcy.

3.2. THE MODEL 51

Section 3.2.4, p. 63.

Making use of (3.2.4) and (3.2.14), these four cases can be formulated by the

following sets Cj, j = 1, . . . , 4, of which each set is defined contingent on promised

repayments:

Case 1 : neither loan suffices to fully redeem deposits .

C1 = (D, `,RD) : DRD ≥ α1` · (D +WB), DRD ≥ α2(1− `) · (D +WB) ,(3.2.16)

Case 2 : only loan 2 suffices to fully redeem deposits.

C2 = (D, `,RD) : α1` · (D +WB) ≤ DRD ≤ α2(1− `) · (D +WB) ,(3.2.17)

Case 3 : only loan 1 suffices to fully redeem deposits.

C3 = (D, `,RD) : α2(1− `) · (D +WB) ≤ DRD ≤ α1` · (D +WB) ,(3.2.18)

Case 4 : either loan suffices to fully redeem deposits.

C4 = (D, `,RD) : DRD ≤ α1` · (D +WB), DRD ≤ α2(1− `) · (D +WB) .(3.2.19)

We will refer to these sets henceforth as “Cases” with a capital letter. These Cases

can be explained as follows: Case 1 represents those structures of the bank’s balance

sheet that require both loans to be fully redeemed in order to be able to pay-off the

depositor as promised. Case 2 represents those structures where the bank can fully

pay-off its depositor if exclusively Loan 2 is fully redeemed, but cannot do so, if

exclusively Loan 1 is fully paid off. Case 3 is the mirror image to Case 2 and Case

4 subsumes those structures where any single loan redemption is sufficient to fully

redeem deposits.5

None of the Cases (3.2.16) to (3.2.19) can be excluded from equilibrium ex ante.

5It can be shown by complete induction that there are 2n+1−n−2 such Cases to be considered ifthere are n loans on the bank’s balance sheet matched to a single debt instrument. This still holdstrue if one out of the n loans is the safe asset as the number of such cases is only determined bythe number of relations between promised redemptions that allow for a full redemption of deposits.Particularly, if the bank potentially held three assets, we would have to consider eleven distinctrepayment schedules for deposits. In particular, holding a volume of the risk-free asset equal toor in excess of the deposit volume does not increase the bank’s expected final wealth compared toautarky without deposit finance.


0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

1.06

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.12

1.14

1.16

1.18

1.20

1.10

1.08

1.06

Case j = 1

j = 2 j = 3

Case j = 1

j=3j = 2

Case j = 2 Case j = 3

j=1

j=4

RD

WB = 0 WB = 100 WB = 1,000

Loan-allocation rate

D

1.10

1.08

1.16

1.18

1.20

1.12

1.14

R

Figure 3.1: The Case constraints given a fixed deposit volume

This figure illustrates the feasible `-RD combinations given a fixed deposit volume, D = 1157.59,and three different levels of the bank’s initial equity WB. Feasible `-RD tuples are separatedaccording to their associated Cases j, j = 1, . . . , 4. The vertical line represents ` ≡ α2

α1+α2.

Parameter values are given according to Table 4.1. The chosen deposit volume of D = 1157.59equals its respective laissez-faire equilibrium value given that WB = 100.

Depending on the parameters, each of these Cases may turn out to be optimal for

the bank. Note that not every loan-allocation rate ` ∈ [0, 1] is feasible given a Case

j. Under Case 2, the loan-allocation rate ` may only range between

` ∈ [ 0,α2

α1 + α2

] (3.2.20)

and under Case 3 between

` ∈ [α2

α1 + α2

, 1] . (3.2.21)

These bounds hold independent of specific values considered for the variables D and

RD as long as the total capital/loan volume is strictly positive, D +WB > 0.

Figure 3.1 displays feasible `-RD combinations for a fixed amount of deposits and

different levels of the bank’s initial equity. The `-RD tuples are separated according

to their associated Cases.

Now, the state-dependent pay-offs arising from the deposit contract will be examined

such that expected pay-offs and the variance of pay-offs can be determined given a

Case j where Dj denotes the stochastic pay-off given Case j and where

D (3.2.22)

3.2. THE MODEL 53

denotes the stochastic pay-off from the deposit contract without referring to any

specific case.

According to the definition of

Case 1 (neither loan suffices to fully redeem deposits)

the realizations of the firms’ success variables (X1, X2) result in the following pay-off

structure:

D1 =

min DRD, α1L1 + α2L2 = DRD, if X1 = 1, X2 = 1

min DRD, α1L1 = α1L1, if X1 = 1, X2 = 0

min DRD, α2L2 = α2L2, if X1 = 0, X2 = 1

min DRD, 0 = 0, if X1 = 0, X2 = 0

(3.2.23)

The expected repayment of deposits is

E(D1) = q ·DRD + [ (p1 − q)α1`+ (p2 − q)α2(1− `) ] · (D +WB)

= µ1 · (D +WB) − q ·WBRD,(3.2.24)

where µ1 is defined as

µ1 = qRD + (p1 − q)α1` + (p2 − q)α2(1− `), (3.2.25)

and represents the marginal expected gross return on deposits.

The variance of the repayments, D1, can be stated as

V(D1) = σ21 · (D +WB)2 + 2 · qRD · (µ1 −RD) ·D ·WB +

+ qRD · [2µ1 − (1 + q)RD] ·W 2B

(3.2.26)

where

σ1 =

√qR2

D + (p1 − q)(α1`)2 + (p2 − q) (α2(1− `))2 − µ21 (3.2.27)

is the volatility associated with the gross return obtained on deposits given in Case

1.

According to

Case 2 (only Loan 2 suffices to fully redeem deposits)

the stochastic repayment, D2, differs from that obtained under Case 1, D1, only if


X1 = 0 and X2 = 1 emerge. This event results in

D2 = min DRD;α2L2 = DRD if X1 = 0, X2 = 1 . (3.2.28)

The expected value of D2 is given by

E(D2) = p2 ·DRD + (p1 − q)α1` · (D +WB)

= µ2 · (D +WB) − p2 ·WBRD

(3.2.29)

where µ2 is

µ2 = p2RD + (p1 − q)α1` . (3.2.30)

D2 has as variance

V(D2) = σ22 · (D +WB)2 + 2 · p2RD · (µ2 −RD) ·D ·WB +

+ p2RD · [ 2µ2 − (1 + p2)RD ] ·W 2B

(3.2.31)

where the volatility of gross deposit returns, σ2, is

σ2 =√p2R2

D + (p1 − q)(α1`)2 − µ22 . (3.2.32)

The deposit repayments given


are symmetric to Case 2 and we obtain

E(D3) = p1 ·DRD + (p2 − q)α2(1− `) · (D +WB)

= µ3 · (D +WB) − p1 ·WBRD

(3.2.33)

as expected pay-off and

V(D3) = σ23 · (D +WB)2 + 2 · p1RD · (µ3 −RD) ·D ·WB +

+ p1RD · [ 2µ3 − (1 + p1)RD ] ·W 2B

(3.2.34)

as the respective variance. The expected gross return on deposits and the associated

volatility are

µ3 = p1RD + (p2 − q)α2(1− `), (3.2.35)

σ3 =√p1R2

D + (p2 − q)α22(1− `)2 − µ2

3 . (3.2.36)

Under

3.2. THE MODEL 55

Case 4 (either loan suffices to fully redeem deposits)

the depositor receives the promised redemption DRD as long as at least one firm is

successful. Hence, the expected pay-off and its variance are:

E(D4) = (p1 + p2 − q) ·DRD = µ4 ·D, (3.2.37)

V(D4) = σ24 ·D2, (3.2.38)

where

µ4 = (p1 + p2 − q)RD, (3.2.39)

σ4 =√

(p1 + p2 − q)R2D − µ2

4 . (3.2.40)

The expected repayments of deposits and its associated variance can be displayed

in one formula across all four Cases if we introduce a new symbol for the Case-

dependent probability that deposits are fully repaid:

qj = P(Dj = D ·RD) =

q, in Case j = 1,

p2, in Case j = 2,

p1, in Case j = 3,

p1 + p2 − q, in Case j = 4.

(3.2.41)

Their interpretation is two-fold: from the depositor’s point of view, qj is primarily

the probability that deposits are redeemed as contracted. From the bank’s point

of view, qj is primarily its survival probability. 1− qj states the bank’s probability

of bankruptcy. The probability qj crucially depends on the composition of the loan

portfolio and the relation of promised loan pay-offs to promised deposit. This issue

will be addressed in detail in this chapter from Section 3.2.4 on.

The expected repayment of deposits thus becomes

E(Dj) = µj ·D + (µj − qjRD) ·WB , j = 1, . . . , 4 , (3.2.42)

and the variance of the repayments can be stated as

V(Dj) = σ2j · (D +WB)2 + 2 · qjRD · (µj −RD) ·D ·WB +

+ qjRD · [2µj − (1 + qj)RD] ·W 2B , j = 1, . . . , 4 .

(3.2.43)


3.2.3 The Household

The household allocates its initial wealth, WH between risky bank deposits,

promising D · RD, and risk-free assets yielding the exogenous gross interest rate

Rf ≥ 1. The household is not allowed to short-sell any assets or to borrow.

It maximizes its final wealth, WH , according to a µ-σ-preference function. This

maximization problem can be considered as a simplified version of a household’s one-

period savings-consumption decision where consumption is not explicitly modeled.

As there is no asymmetric information, the depositor knows the possible portfolio

compositions that are feasible for the bank and values the offered deposit contracts

according to their risks and possible returns. Likewise, the bank cannot cheat and

commits to each contract.6

Final wealth equals under any of the four Cases

WH = D + (WH −D)Rf . (3.2.44)

As the distribution of D changes along changing triples (D, `,RD), so does the

distribution of final wealth WH . In particular, if the variables ` and RD are fixed,

the distributions of D and WH solely depend on D.

The µ-σ-preferences over final wealth, WH , are

U(WH) = E(D) + (WH −D) ·Rf −1

2· γ · V(D) . (3.2.45)

The parameter γ > 0 reflects the household’s degree of risk-aversion. Given Case j,

the utility U(WH) is characterized by the expected deposit pay-off E(Dj) and the

associated variance V(Dj), as given by (3.2.42) and (3.2.43).

6Dermine (1986), Calem/Rob (1999), and Zhu (2008) also study bank models with uninsureddeposits, but, contrary to our model, depositors are risk-neutral. In Calem/Rob (1999, p. 344),the depositor takes the bank’s initial equity and the bank’s portfolio-allocation rate into account.But Calem/Rob need not consider different deposit repayment schedules as they simply considerredemptions from the bank’s loan portfolio on the return level and split them into safe returnsand risky returns modeled by a uniform distribution. Also Dermine (1986) allows the bank tohold a risky commercial loan and a riskless bond, but he does not consider the case that the bondredemption could exceed the promised deposit redemption. In Zhu (2008, p. 178), the bank has onlyone asset that can be affected by shocks and the depositor internalizes the bank-specific shock only(Zhu, 2008, p. 178, Eq. (5)). Hence, all three papers have in common, that agents deal with a singlesource of risk. Calem/Rob (1999, p. 346) further question the depositors’ abilities of observing theallocation rate and note that related empirical evidence is lacking. We note that, in reality, thecommitment problem may be a problem of both asymmetric information and timing, i.e. the bankwill reveal its asset choice at most when its financing is fixed (Bhattacharya/Boot/Thakor, 1998,p. 756).

3.2. THE MODEL 57

Given a fixed loan-allocation rate `, a fixed deposit interest rate RD, and a fixed

Case j, the household chooses that amount of deposits D that maximizes its utility

as stated in (3.2.45) at which the optimal deposit volume must be from [0,WH ].

Therefore, the household’s optimization problem is given by

maxD

U(WH),

s.t. `, RD, j fixed , (3.2.46)

0 ≤ D ≤ WH .

The unconstrained optimization problem does not account for the lower and the

upper bound on D, 0 ≤ D ≤ WH , and will be referred to by (3.2.46’). As U(WH) is

strictly concave in D given a specific Case j, the solution to Problem (3.2.46’)

is unique within each Case j and denoted by Du(`, RD; j). Du(`, RD; j) is the

unconstrained deposit-supply function. In general, magnitudes referring to Problem

(3.2.46’) or to its solution Du(`, RD; j) will be super-indexed by u. U(WH) is strictly

concave in D within each Case j, as

∂2U(WH)

∂D2= − γσ2

j , j = 1, . . . , 4 , j fixed ,

holds.

Solutions to Problem (3.2.46’) are derived for given values of ` and of RD, and

for a given Case j by taking, for each of the four Cases separately, the first partial

derivative of U(WH) with respect to D for fixed ` and RD and solving the associated

first-order condition. That is, the functions Du(`, RD; j) are Case-wise maximizers

to Problem (3.2.46’).

We dispense with the constraints associated with Cases 1 to 4 for the following

reason: as the bank is a monopolist on both the loan and the deposit market,

it will choose its risk-return profile such that its objective (i.e. expected wealth)

is maximized. This objective is reached by choosing the appropriate loan-

allocation rate ` and deposit interest rate RD within a given Case j such that

the solution is feasible in the sense that it meets one of the respective constraints

as given by (3.2.16) to (3.2.19). Doing so for each of the four Cases results in

Case-wise optimal choices which will be denoted by (`∗(j), R∗D(j)). The choices

(`∗(j), R∗D(j)), Ds(`∗(j), R∗D(j); j)) must meet the definition of the respective set

Cj, as defined by (3.2.16) to (3.2.19), respectively. The triple resulting in the

highest expected wealth is the final solution. For details, we refer to Section 3.3


and Subsection 3.3.1, respectively.

As a consequence, we can analyze the household’s decision problem given each of the

four Cases j for any fixed pair of ` and RD, i.e. we need not care if a given solution

to the household’s Problem (3.2.46) under one Case given ` and RD dominates a

solution under another Case given the same values of ` and RD. Instead, that deposit

volume and its associated Case will prevail in equilibrium that leads to the highest

bank’s expected wealth given a fixed pair of ` and RD.

Therefore, we can separately derive the explicit solutions to Problem (3.2.46’)

and thus for Problem (3.2.46). The solution to Problem (3.2.46) is denoted by

Ds(`, RD; j) and will be referred to as the constrained solution.

Result 1 states the formulæ for the household’s deposit-supply function and provides

some of its basic properties.7

Result 1. The unconstrained deposit supply Du(·) that maximizes Problem (3.2.46)

is unique under each Case j, j = 1, . . . , 4, and given by

Du(`, RD; j) =µj −Rf

γσ2j

+qjRD(RD − µj)− σ2

j

σ2j

·WB, j = 1, . . . , 4 . (3.2.47)

Under Case 4 or if the bank grants only one loan in Cases 2 or 3, the unconstrained

supply function simplifies to:

Du(`, RD; j) =µj −Rf

γσ2j

, for either j = 4, or j = 2, ` = 0, or j = 3, ` = 1 .

(3.2.48)

The constrained deposit-supply function reads

Ds(`, RD; j) = min max Du(`, RD; j), 0 ;WH , j = 1, . . . , 4 . (3.2.49)

In each Case j, j = 1, . . . , 4, the functions Du(`, RD; j) and Ds(`, RD; j) are

continuous and Du(`, RD; j) is differentiable both in its variables, ` and RD, and

in all its parameters.

As the deposit-supply functions Ds(`, RD; j) are optimal decisions given a fixed Case

j, the optimal deposit supply can migrate from one Case to another through changes

in the variables ` and RD or through changes in parameter values.

In Case 4, the deposit-supply function is independent of the bank’s equity, WB, for

the following reason: the depositor receives the promised payment D ·RD if at least

7A version of this result can be found in Buhler/Koziol/Sygusch (2008, p. 142).

3.2. THE MODEL 59

one loan is redeemed and nothing if both firms fail. Thus, the contingent repayments

of the deposits are independent of the bank’s initial equity WB under Case 4 and so

is the deposit-supply function Du(`, RD; 4). For the same reason the productivity

parameters (optimal gross rates on the firms’ debts) α1 and α2 do not enter the

deposit-supply function under Case 4. We can argue analogously if the bank grants

a single loan only given Case 2 and 3, respectively.

Therefore, the deposit-supply function depends on the initial equity WB under

the Cases where WB is crucial for the residual loan portfolio value if the bank

defaults. To be more precise, the dependence on WB emerges whenever two loans

are effectively granted under Cases 1, 2, and 3.

The first term of (3.2.47), apart from the composition of µj and σj, represents the

optimal allocation of funds if an investor with µ-σ-preferences can choose between

a risky asset and a risk-free asset whose pay-offs are linked linearly to the amount

of funds invested.

If, furthermore, the pay-off from the risky asset depended linearly on funds WB

invested by another party, say a bank, this very investor would supply funds equal

toµj −Rf

γσ2− WB , (3.2.50)

as long as the investor has still the opportunity to store funds into the risk-free asset.

The funds invested by the other party/the bank lower ceteris paribus the supply of

funds by the µ-σ investor as any gains by the own funds invested is enhanced by

WB.

Finally, the termqjRD(RD − µj)

σ2j

·WB

reflects the non-linearity in the marginal gains and the marginal risk of the deposit

contract.

Next, Result 2 will characterize the household’s deposit-supply function dependent

on the deposit interest rate RD. Furthermore, Result 2 will be the basis on which

the bank’s Optimization Problem (3.3.1) will be analyzed, as it provides an upper

bound to RD.

Therefore, let us define the following critical interest rates on deposits: RD(`; j)

denotes the critical deposit interest rate where the unconstrained deposit-supply

function Du(`, RD; j) becomes zero (given a fixed loan-allocation rate ` and given a


fixed Case j):

RD(`; j) ∈ RD| Du(`, RD; j) = 0 , (3.2.51)

Ru

D(`; j) is the deposit interest rate that maximizes the unconstrained deposit-supply

function Du(`, RD; j) (given a fixed loan-allocation rate ` and given a fixed Case j):

Ru

D(`; j) ∈RD| Du(`, R

u

D(`; j); j) ≥ Du(`, RD; j) ∀ RD ≥ RD(`; j). (3.2.52)

RD(`; j) represents the lowest deposit interest rate that maximizes the (constrained)

deposit-supply function Ds(`, RD; j):

RD(`; j) := min

Ru

D(`; j), RD|RD = infRDDs(`, RD; j) = WH

.

Thus, RD(`; j) ≤ RD(`; j) ≤ Ru

D(`; j) holds true by definition if Ru

D(`; j) exists

where equality for the latter relation means that the household’s deposit supply is

not restricted by its initial wealth. The former relation is trivial.

The unconstrained deposit-supply function shows the following behavior in RD. The

existence of RD(`; j) and RD(`; j) has been also put forward by Buhler/Koziol/Sy-

gusch (2008, p. 143). Here, we also provide a proof that can be found in Appendix

A.1.1.

Result 2. For each Case j and for ` fixed, exactly one interest rate RD(`; j)

exists for which the unconstrained deposit-supply function becomes zero, and exactly

one interest rate Ru

D(`; j) exists that maximizes the unconstrained deposit-supply

function with respect to RD. The unconstrained deposit-supply function approaches

asymptotically zero from above if RD goes to infinity given any of the four

Cases. Given Case j, the unconstrained deposit-supply function increases strictly

monotonically in RD on [RD(`; j), Ru

D(`; j)] and decreases strictly monotonically on

[Ru

D(`; j),∞).

Briefly, the unconstrained deposit-supply function is hump-shaped in RD on

[RD(`; j),∞) given Case j, that is deposit supply is backward bending.

Clearly, Result 2 can be extended to the case of the constrained deposit-supply

function which is zero for all RD ≤ RD(`; j) and potentially flat for RD on

some interval if the initial wealth WH is lower than the maximum value of the

unconstrained supply function.

The economic intuition behind Result 2 can be understood by the pay-off structure

3.2. THE MODEL 61

of the standard risky debt contract and by the investor’s risk-aversion level: The

higher the promised interest rate RD grows, the higher the promised pay-off D ·RD

becomes. Thus, it is natural to increase the supply of funds D in order to further

increase the final promised pay-off. However, deposits are risky and the household

is risk-averse. Therefore, there is a point at which the benefits of further boosting

D do not outweigh the associated costs in terms of a rising variance. Consequently,

the household will lower D if the deposit interest rate RD is further increased.

Specifically, the promised pay-off D ·RD may still rise while the household deposits

less and less funds. Ultimately, if RD approaches infinity, deposit supply falls to

zero.

Finally, let us characterize the critical interest rate RD(`; j). We define the following

Case-dependent magnitudes

Rµj = µj − qjRD

Rσj = σ2

j + µ2j − qjR2

D

, j = 1, . . . , 4, `, j fixed, (3.2.53)

where Rµj denotes the marginal expected gross return on deposits exclusively the

state of full deposit redemption and Rσj is the associated second, non-centered

moment. As a consequence,

Rσj − (Rµ

j )2 ≥ 0 (3.2.54)

is a variance and thus positive. Rµj and Rσ

j are independent of RD. In Case j = 4,

both, Rµ4 and Rσ

4 , become zero, as the bank defaults if and only if both firms default.

Result 3. The interest rate RD(`; j) for which the unconstrained deposit supply

function becomes zero is given by

RD(`; j) =Rf −Rµ

j + γWB ·[Rσj − (Rµ

j )2]

qj · (1 + γWB ·Rµj )

, ` fixed, j = 1, . . . , 4 . (3.2.55)

It decreases strictly in WB in Cases 2 and 3. In Case 4, the interest rate RD(`; 4) is

independent of WB due to Rµ4 = Rσ

4 = 0. RD(`; 4) always exceeds Rf and simplifies

to

RD(4) =Rf

q4

.

If WB = 0,

RD(`; j) =Rf −Rµ

j

qj> Rf , ` fixed, j = 2, 3,


holds in Cases 2 and 3. The condition

αi <1− qpi − q

·Rf , i = 1, 2 , (3.2.56)

is sufficient to obtain RD(`; 1) > Rf in Case 1 if WB = 0.

The critical interest rate RD(`; j) remains positive, if WB goes to infinity,

limWB→∞

RD(`; j) =Rσj − (Rµ

j )2

qj ·Rµj

≥ 0, ` fixed, j = 1, 2, 3 .

The relation RD(`; j) > Rf holds in WB = 0 since the household is only willing to

bear the credit risk of the deposits if they yield a higher expected return than the

risk-free rate Rf .8 The higher is the bank’s initial equity WB, the lower the credit

spread on deposits becomes, as the buffer function of the bank’s equity gains in

importance. In particular, RD(`; j) strictly decreases in WB in the Cases j = 2, 3.

In Case 1, this relation generally remains opaque, however. In Case 4 or if loan-

allocation rates are either zero (Case 2) or one (Case 3), contingent payments from

the deposit contract are not affected by WB, and so is RD(`; j) (cf. Result 1 and

Formulæ (3.2.24), (3.2.29), (3.2.33), and (3.2.37)).

The critical interest rate RD(`; j), j = 1, 2, 3, is negatively related to WB in such

a way that it may even fall below Rf for sufficiently high amounts of bank capital.

The limit of RD(`; j) if WB goes to infinity marks the bottom line of which the

positive sign is guaranteed by (3.2.54).

The following result illustrates that the buffer function of the bank’s initial equity

WB may stipulate the deposit supply.

Result 4. Let µj ≥ Rf and furthermore, let ` 6= 0 in Case 2, and ` 6= 1 in Case 3.

Then,∂Du(`, RD; j)

∂WB

> 0 (3.2.57)

holds for the Cases j = 2 and 3.

Let us close this section by discussing the household’s deposit supply in Case 4.

Remark 1. The characteristics of deposit supply in Case 4 can be stated in closed

8We note that the additional (and only sufficient) bounds in Case 1 are rather of the sameorder of magnitude as the bounds assumed in (3.2.7). The base case scenario, as given in Table4.1, yields as bounds according to (3.2.56) 1.70625 for α1, and 4.55 for α2. Assumption (3.2.56) isnot further considered in what follows.

3.2. THE MODEL 63

form. The unconstrained deposit supply, Du(`, RD; 4), becomes zero at

RD(4) =1

q4

·Rf ,

and attains its unique maximum at

Ru

D(4) =2

q4

·Rf ,

as Du(`, RD; 4) is strictly increasing for all RD < 2q4Rf and strictly decreasing

otherwise. If deposit supply is constrained by WH , the critical rate RD(`; j) where

Du(`, RD; 4) just attains WH is

RD(4) =q4 −

√q2

4 − 4γq4(1− q4)WHRf

2γq4(1− q4)WH

with

limWH↓0

RD(4) = RD(4) .

3.2.4 The Bank’s Expected Pay-off

The bank is a risk-neutral intermediary between the firms and the household and

has limited liability.9 It exerts monopoly power on both markets: the one for loans

and the one for deposits. As a result, the bank charges

R∗i = αi

as gross interest rate on loans while being able to scale the loan volumes arbitrarily

and it takes the household’s (constrained) deposit-supply function as given when it

maximizes its expected final wealth. Deposit supply differs according to the four

Cases defined in (3.2.16) to (3.2.19) as the bank’s probabilities of solvency differ

as well. Given a Case j, the bank does creditably commit to this Case. Thus the

bank does not arbitrarily combine deposit supply schedules with its own repayment

schedules.

According to

Case 1 (neither loan suffices to fully redeem deposits),

defined by (3.2.16), the bank remains only solvent if both loans are fully repaid which

9The following notes and formulæ in this section can be found on pp. 144f in Buhler/Koziol/Sy-gusch (2008).


happens with probability q. If only one single firm succeeds, the bank must cede the

associated loan redemption fully to the depositor. With probability 1− p1− p2 + q,

there is no payment flow. Thus, using the loan-allocation rate defined by (3.2.14),

the bank’s objective function takes the following form under Case 1:

E[WB(`, RD; 1)

]= q · [ α1`+ α2(1− `)−RD ] ·Ds(`, RD; 1)

+ q · [ α1`+ α2(1− `) ] ·WB .(3.2.58)

According to

Case 2 (only Loan 2 suffices to fully redeem deposits),

defined by (3.2.17), the bank keeps solvent if both loans are fully redeemed or if the

loan L2 is fully paid back:

E[WB(`, RD; 2)

]= [ qα1`+ p2α2(1− `)− p2RD ] ·Ds(`, RD; 2)

+ [ qα1`+ p2α2(1− `) ] ·WB .(3.2.59)

Symmetrical to Case 2,


results in the following objective function:

E[WB(`, RD; 3)

]= [ p1α1`+ qα2(1− `)− p1RD ] ·Ds(`, RD; 3)

+ [ p1α1`+ qα2(1− `) ] ·WB .(3.2.60)

Finally, the bank keeps solvent as long as at least a single loan is fully paid back if

it lends according to

Case 4 (either loan suffices to fully redeem deposits):

E[WB(`, RD; 4)

]= [ p1α1`+ p2α2(1− `)− (p1 + p2 − q)RD ] ·Ds(RD; 4)

+ [ p1α1`+ p2α2(1− `) ] ·WB .

(3.2.61)

Case 4 is the only case which results in an objective function that is linear in the loan-

allocation rate `. Thus, if Case 4 is feasible, the bank always chooses an allocation

rate at the corner of the feasibility set C4 given by (3.2.19). This idea can be further

developed both with and without regulation.

Results 9 and 11 will characterize Case 4 as laissez-faire equilibrium. Results 19

3.3. EQUILIBRIUM AND SENSITIVITIES 65

and 21 refer to Case 4 as equilibrium under regulation by fixed risk weights, and

finally Results 24 and 28 given a VaR approach. Case 4 can arise as equilibrium

both under equal and under different projects the two firms undertake.

3.3 Equilibrium and Sensitivities

3.3.1 The Equilibrium without Regulation

3.3.1.1 General Characterization

Given each Case j, the bank maximizes its expected final wealth, E[WB(`, RD; j)],

with respect to the loan-allocation rate ` and the interest rate on deposits RD.10

The tuple (`∗(j), R∗D(j)) denotes the allocation rate and the interest rate that

maximize E[WB(`, RD; j)

]under Case j. The triple (`∗, R∗D; j∗) denotes the loan-

allocation rate, the interest rate on deposits, and the associated Case that maximize

E[WB(`, RD; j)

]considering all the four Cases, i.e. (`∗, R∗D; j∗) is the solution to

the following maximization problem:11

max`, RD

E[WB(`, RD; j)

]

s.t. ` ∈ [0, 1] (3.3.1)

∃j : (Ds(`, RD; j), `, RD) ∈ Cj j = 1, . . . , 4 .

We will refer to (`∗, R∗D; j∗) as the equilibrium without regulation. Equilibrium

results are used later on as a benchmark to judge potential pro-cyclical impacts

resulting from regulation.

Result 5. An equilibrium (`∗, R∗D; j∗) always exists. The optimal interest rate on

deposits given Case j, R∗D(`; j), satisfies R∗D(`; j) ∈(RD(`; j), RD(`; j)

], and is

unique given a fixed loan-allocation rate `. Hence the optimal deposit volume D∗ ≡Ds(`∗, R∗D; j∗) is always strictly positive.

The existence of an optimizing triple (`∗, R∗D; j∗) holds for two reasons: the sets Cj

defining the Cases j = 1, . . . , 4 are compact sets by definition. As a consequence,

we obtain compact sets with respect to the loan-allocation rate ` given fixed deposit

10Observe that the optimal interest rates on loans are simply Ri = αi, i = 1, 2, independent ofthe Case j considered, cf. (3.2.4).

11Buhler/Koziol/Sygusch (2008, p. 145, Maximization Problem (16)).


interest rates RD. Thus, bounds on ` are established that are tighter than the short-

sell restrictions, as outlined for Cases 2 and 3 by (3.2.20) and (3.2.21), respectively.

The borders arising out of Cases 1 and 4 cannot be analytically stated. Concerning

the deposit rate RD, Result 2 helps by providing definite bounds. Furthermore,

we can even show the uniqueness of R∗D(`; j). The proof is provided in Appendix

A.2.1.1.12

Optimal solutions (`∗, R∗D; j∗) typically consist of solutions in the interior concerning

the loan-allocation rate, `, as any optimum is a compromise on risky lending between

risk-neutral bank owners and risk-averse depositors. If the risk-neutral bank granted

only its own capital WB as loans, it would always choose to grant a loan only to

the company yielding the highest expected return (by (3.2.5), the second firm). The

same holds true if the bank issues risk-free deposits where risks are assumed by an

exogenous agency (deposit insurer) that charges the bank a fee which is independent

of the risks taken by the bank. As deposits are uninsured and as depositors are risk-

averse, there is an incentive, even for the bank acting as a monopolist on the market

for deposits, to diversify risks.

By the parameters considered in Chapter 4, it will turn out that the unregulated

bank mainly grants loans such that Case 1 prevails. Unregulated equilibria according

to Cases 3 and 4 do not arise in the examples presented in this section. Thus, we

find the typical risk-reducing effect through uninsured debt as long as both parties

are fully informed about each other. So, in our framework, uninsured deposits can

fully unfold their disciplinary effects. This does not imply, however, that there is

no scope for choosing extreme loan-allocation rates, such as `∗ = 0 or `∗ = 1 given

some parameter values. Figure 3.2 shows feasible `-RD combinations under the base

case scenario.

If `∗ = 0 holds, the equilibrium is characterized by Case 2. Likewise, `∗ = 1 is always

associated with Case 3.

In Case 1, loan-allocation rates equal to `∗ = 0 or `∗ = 1 would result in R∗D > α2,

and R∗D > α1, respectively, implying (due to Result 2) a negative contribution of

deposit-taking to the final wealth expected by the bank. This contradicts optimality,

as the bank can improve by changing to Case 2, or 3, respectively. In Case 4, choices

`∗ = 0 and `∗ = 1 contradict D∗ > 0, which also prevails as a maximum under this

Case. Furthermore, choosing `∗ = 0 does not generally dominate `∗ = 1 under Case

3 even if p2α2 > p1α1 is assumed, because the bank promises different interest rates

12Existence and bounds on R∗D have been also formulated by Buhler/Koziol/Sygusch (2008,

p. 145), without providing the proof, however.


0.0 0.2 0.4 0.6 0.8 1.0

1.06

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.12

1.14

1.16

1.18

1.20

1.10

1.08

1.06

Case j = 1

j= 2 j = 3

Case j = 1

j = 3j = 2

Case j = 2 Case j = 3

j=4

RD

WB = 0 WB = 100 WB = 1,000

Case j = 1

Loan-allocation rate

D

1.10

1.08

1.16

1.18

1.20

1.12

1.14

R

Figure 3.2: The Case constraints given the household’s deposit-supplyfunctionThis figure illustrates the feasible `-RD combinations given the household’s deposit supply,Ds(`, RD, j) for three different levels of the bank’s initial equity WB. Feasible `-RD tuples areclustered by their associated Cases j, j = 1, 2, 3, 4. The vertical line represents ` ≡ α2

α1+α2.

Parameter values are given according to Table 4.1. Cf. Figure 3.1 where the deposit volume isexogenously fixed to D = 1157.59.

R∗D to the depositor under both Cases. The solution `∗ = 0 is discussed in depth in

the next section.

Moreover, Result 5 shows that it is always worthwhile for the bank to grant loans,

even if it has no initial equity. So the bank benefits from its role as a financial

intermediary that is assumed to have exclusive access to investment in the firms’

projects.

In the next sections, we will analyze some specific optimal choices of the bank which

can be potential equilibria. In general, however, there is no explicit characterization

of these choices possible for Cases 1 to 3. For Case 1, we only consider the case of

two equally distributed projects, for Case 2 the corner solution in `∗ = 0.

3.3.1.2 Characterization if Case 1 Prevails

Contrary to Assumption (3.2.5), let us consider equal success probabilities p1 = p2 =

p and equal productivity rates α1 = α2 = α. Hence, both loans have equal expected

return and variance. The optimum given Case 1 can be stated analytically. However,

the comparative static properties under Case 1 remain opaque. The following result

summarizes some properties of this potential equilibrium:

Result 6. Let p1 = p2 = p and α1 = α2 = α. Suppose Case 1 is optimal. Then the


bank allocates its funds equally as loans to both firms,

`∗ =1

2,

and promises

R∗D =qα·[2(1−q)Rf − (3−2p)(p−q)α + (1−p)(p−q)(1−2p+2q)α2γWB] + α·

√Q∗1(p,q,α)

2·q·(1−q)Rf + [2q−p(1+q)]α + 12

(p−q)(1−2p+3q−2pq)α2γWB

as the gross interest rate to the household where

Q∗1(p, q, α) = q · [p(1− p) + q − p2] ··

2(1− q)R2f − 4(p− q)αRf + (2p− q)(p− q)α2 + . . .

+ 12(p− q)(1− 2p+ q)α2γWB [4 (Rf − (p− q)α) + (1− 2p+ 2q)(p− q)α2γWB]

.

This maximum choice is unique given Case 1.

Furthermore, one can specify the optimal deposit volume and the bank’s optimal

expected wealth but their analytical representations are not very meaningful. By

the equality of both loan distributions there is no disaccord among the risk-neutral

bank and the risk-averse depositor about the optimal loan-allocation rate `∗ in Case

1. But the interest rate R∗D, that the bank promises to its depositor, is still lower

than that preferred by the household as long as R∗D < Ru

D(`; 1) holds, cf. Result 5.

The derivation of this result can be found in Appendix A.2.1.2.

The sign of the derivative of R∗D with respect to WB is ambiguous and therefore it

is unclear whether the marginal change of the total loan volume towards a marginal

change in the bank capital is higher or lower than one and even whether this marginal

rate of change is positive or not. But there is a sufficient condition on α such that

the slope of D∗ is strictly positive with respect to WB:

Result 7. Let p1 = p2 = p and α1 = α2 = α. Assume Case 1 is optimal and α ≤ 43.

Then,dD∗

dWB

> 0

holds.

The additional restriction on α, given by α ≤ 43, instead of the general Assumption

(3.2.7), is only sufficient to obtain the result. Hence, there are optimal choices of

the bank under Case 1 with α > 43

and dD∗

dWB> 0. The proof to this result is outlined

in Appendix A.2.1.3.


3.3.1.3 Characterization if Case 2 with `∗ = 0 Prevails

In this section, we characterize Case 2 when the optimal loan-allocation rate `∗ is

chosen to be zero. The following result outlines the sufficient conditions.

Result 8. Suppose that the household is not constrained by its initial wealth, WH ,

and that, given Case 2, the expected final wealth of the bank, E[WB(`, RD; 2)], has a

unique maximum in ` on R. Furthermore, assume

(2− p2)(qα1 − p2α2)Rf + p2(p2α2 − p1α1)Rf + p22(p1 − q)α1α2 ≤ 0 , (3.3.2)

and

(p22α

22 −R2

f ) · [(q − p1p2)α1Rf − (1− p2)p2α2Rf + (p1 − q)α1p2α2] −− 2p2(1− p2)α2

2γWBRf ·· (2− p2)(p2α2 − qα1)Rf − p2(p2α2 − p1α1)Rf − p2

2(p1 − q)α1α2 ≤ 0

(3.3.3)

hold, then

`∗ = 0 and R∗D =2α2Rf

p2α2 +Rf

> Rf

are optimal given Case 2. The sensitivities of the total loan volume, L∗, and the

deposit volume, D∗, to shocks in bank equity WB simplify to

dL∗

dWB

= 1 anddD∗

dWB

= 0 . (3.3.4)

Condition (3.3.3) guarantees that `∗ = 0 is the optimal choice given that there is

a unique maximum of E[WB(`, RD; 2)] for fixed RD. Condition (3.3.2) determines

exclusively the sign of the second derivative ∂2E[·]∂WB∂`

such that the optimal loan-

allocation rate ` shrinks with increasing bank capital if short-sells of bank assets

were allowed. Thus, the constrained loan-allocation rate, i.e. the loan-allocation

rate ` restricted to the unit interval, remains zero for increasing WB.

As `∗ is kept fixed at zero, the optimal deposit interest rate R∗D is also constant with

respect to WB, such that only the direct effect of a marginal change in WB affects

the total loan volume L∗. Note that in Equation (3.3.4) the direct partial derivative

of deposits, D∗, with respect to bank capital, WB, vanishes according to (3.2.48).

Appendix A.2.1.4 provides the derivation of these results.

The importance of the Conditions (3.3.2) and (3.3.3) can also be grasped from a

rather economic perspective: the left-hand side of each of the conditions becomes

smaller, as the gap between both loans’ marginal expected gross returns grows. This


gap is measured in terms of p2α2− p1α1 and p2α2− qα1, respectively. In particular,

the negative of the first two terms in (3.3.2)

(2− p2)︸︷︷︸>p2

(p2α2 − qα1)︸︷︷︸>(p2α2−p1α1)

Rf − p2(p2α2 − p1α1)Rf

is positive due to (3.2.5). Thus, both the Conditions (3.3.2) and (3.3.3) are

negatively affected.

Condition (3.3.2) excludes

∂2E[WB(`, RD; 2)]

∂` ∂WB

> 0 .

But as soon as the opposite to (3.3.2) holds true and if furthermore13

∂2E[WB(`, RD; 2)]

∂` ∂RD

< 0

is assumed, the sensitivities of the loan-allocation rate `∗ and the deposit interest

rate R∗D show opposite reactions to shocks in WB,

d`∗

dWB

> 0 anddR∗DdWB

< 0 .

Consequently, the effect on the total loan volume remains ambiguous as the first-

order conditions imply that the optimal deposit supply is increasing in both the

loan-allocation rate as well as in the deposit interest rate.

Remark 2. Given that ` = 0 remains optimal for the constraint maximization

problem for changes in the following parameters, the optimal total loan volume reacts

as follows:

dL∗

dp2

=p2(p2α

22 −R2

f ) + (1− p2)R2f

4p22(1− p2)2α2

2Rfγ> 0 (3.3.5)

dL∗

dα2

=Rf

2p2(1− p2)α32γ

> 0 (3.3.6)

dL∗

dγ= −

p22α

22 −R2

f

4p2(1− p2)α22Rfγ

< 0 (3.3.7)

dL∗

dRf

= −p2

2α22 +R2

f

4p2(1− p2)α22R

2fγ

< 0 (3.3.8)

13The opposite does not occur if parameter values are chosen that are close to those given inTable 4.1. In particular, success probabilities are supposed to be close to one.


Table 3.1: Parameter values satisfying Conditions (3.3.2) and (3.3.3)

The upper panel reports parameter values that fulfill Conditions (3.3.2) and (3.3.3) stated in Result8. The lower panel reports the bank’s optimal choices in each Case j = 1, . . . , 4. The equilibriumis characterized by Case 2.

Panel A: Parametrization

p1 p2 q α1 α2 WB γ WH Rf

0.99 0.98 0.975 1.1 1.2 100 0.05 3000 1.05

Panel B: Optimal Choices

`∗(j) R∗D(j) j D∗(·, ·; j) E[WB(·, ·; j)] U(WH) Eq./Res.

0.513879 1.10128 1 112.628 117.187 3153.18 n/a0 1.13208 2 47.3214 120.75 3151.41 Res. 8

0.521739 1.09082 3 111.028 118.084 3152.1 n/a0.39307 1.07471 4 67.3131 119.058 3150.65 (3.3.11)

These formulæ can be obtained by calculation given the (assumed) optimal deposit

volume Du(0, R∗D, 2). The signs are intuitive: either if the risk reduces by an

increasing success probability p2 or if the expected gross return on the second firm

loan, α2, increases, the total loan volume increases too. The total loan volume

decreases if the household’s risk-aversion γ increases. The same effect holds true if

the return on the risk-free sovereign bond, Rf , rises.

Table 3.1 shows a parameter set that fulfills Conditions (3.3.2) and (3.3.3). Thus,

the tuple (`∗, R∗D) = (0, 2α2

p2α2+Rf) is the bank’s optimal choice given Case 2. What is

more, this bank’s choice under Case 2 dominates the choices according to all other

Cases. An equilibrium such as this may also prevail if both firms’ projects exhibit

equal returns and risks as the example provided in Table 3.3 illustrates.

It is worth noting that the bank chooses in each Case the lowest possible loan-

allocation rate, i.e. it devotes the lowest possible amount of funds as loan to Firm

1. Specifically, `∗(3) = α2

α1+α2holds in Case 3, cf. (3.2.21). The solution given Case

4 can also be analytically represented and is given by (3.3.11) in the next result,

Result 9.



If Case 4 is optimal for the bank, the equilibrium can be stated analytically. The

following result describes this equilibrium:

Result 9. Suppose that the household is not constrained by its initial wealth, WH .

Then the maximum (`∗, R∗D, 4) given Case 4 is unique and α1L∗1 = D∗R∗D holds.

Furthermore, if

0 < WB ≤ [p22α2(q4α1−Rf )−(p22−q2)α1Rf ]α1Rf+p22α2(q4α1−Rf )[(α1+α2)Rf−q4α1α2]4p22(1−q4)q4Rfα

21α

22γ

(3.3.9)

holds, the optimal loan-allocation rate and the optimal deposit interest rate are given

by

`∗ = α2

α1+α2,

R∗D =(α1+α2)Rf+q4α1α2+

√[(α1+α2)Rf−q4α1α2]

2+4q4(1−q4)α2

1α22γWBRf

2q4[α1+α2−(1−q4)α1α2γWB ],

with

R∗D > 1q4Rf

Remark 1= RD(4).

(3.3.10)

Consequently, the promised loan redemptions are equal, i.e. α1L∗1 = α2L

∗2. If

Condition (3.3.9) is violated, α1L∗1 < α2L

∗2 prevails implying

`∗ =2p2α2Rfp2α2q4α1−[p2α2+(p2−q)α1]Rf

(p2α2q4α1)2−[p2α2+(p2−q)α1]2R2f+4q4(1−q4)p22α

21α

22γWBRf

∈ (0, α2

α1+α2) ,

R∗D =2p2α1α2Rf

p2α2q4α1+[p2α2+(p2−q)α1]Rf> 1

q4Rf

Remark 1= RD(4) ,

D∗ =p2α2q4α1−[p2α2+(p2−q)α1]Rf·p2α2q4α1+[p2α2+(p2−q)α1]Rf

4(1−q4)q4p22α21α

22γRf

.

(3.3.11)

For WB = 0, the loan-allocation rate `∗(4) is indeterminate. R∗D(4) = 1q4Rf and

D∗(4) = 0 holds. Then Case 4 is strictly dominated by choices according to other

Cases and can thus never arise in equilibrium.

The solution characterized by (3.3.10) and by (3.3.11) is continuous in WB.

The relation α1L∗1 = D∗R∗D is a direct consequence of the assumption p1α1 < p2α2

as the bank forgoes more profits for each marginal unit of funds it grants to Firm

1 more than required by the feasibility set C4. Technically, the equality α1L∗1 =

D∗R∗D determines the optimal interest rate on deposits the bank chooses. As this

choice depends on the loan-allocation rate, either the optimal loan-allocation rate


maximizes the bank’s expected wealth freely while fulfilling the bounds given by the

set C4, i.e. α2L∗2 > D∗R∗D, or the optimal loan-allocation rate must be chosen such

that α2L∗2 = D∗R∗D holds as well. If WB strictly exceeds the threshold (3.3.9), the

solution presented under (3.3.11) arises, in particular, α2L∗2 > D∗R∗D holds. The

derivations of the formulæ are outlined in Appendix A.2.1.5.

The relation between WB and the occurrence of Case 4 in equilibrium

Note that Case 4 can only be optimal for sufficiently high bank capital, WB, or with

sufficiently high productivity parameters αi, i = 1, 2: consider WB = 0 and αi ≤ 2.

Then the definition of Case 4 according to (3.2.19) implies

α1` ·Ds(·, 4) ≥ Ds(·, 4) ·RD and α2(1− `) ·Ds(·, 4) ≥ Ds(·, 4) ·RD .

For Ds(·, 4) > 0, we obtain RD ≤ min α1`, α2(1− `). By αi ≤ 2 and ` ∈ [0, 1],

min α1`, α2(1− `) ≤ 1 holds. Hence, RD ≤ 1, and especially RD ≤ Rf . This

is a contradiction to Ds(·, 4) > 0 as the critical deposit interest rate, at which

deposit supply becomes zero, is RD(4) = 1q4Rf > Rf . Thus, only Ds(·, 4) = 0

is feasible, implying zero expected wealth of the bank. But the bank can simply

improve by choosing a loan-allocation rate compatible with any other Case j =

1, 2, 3. Particularly the choice discussed in Result 8 is feasible under WB = 0 and

thus is a potential candidate for the Maximization Problem (3.3.1). Alternatively,

one can argue by the optimal deposit interest rate R∗D(4) in (3.3.10) if WB lies

beneath the threshold (3.3.9). At WB = 0, R∗D(4) becomes 1q4Rf ≡ RD(4) and we

obtain Ds(`, RD(4); 4) = 0.

Table 3.2 illustrates the relation between the bank’s initial capital WB, the different

optimal choices within Case 4, and the actual optimal choices by the bank across

all Cases, i.e. the equilibria. It suggests a positive relation between the bank’s

initial capital WB and the occurrence of Case 4 in equilibrium. The intuition for

this relation is as follows: the higher WB, the easier for the bank to find loan

allocations that fulfill the Case constraints of Case 4. Specifically, for WB above

(3.3.9), the deposit supply and the deposit interest rates are fixed in equilibrium such

that allocating increasing funds to Firm 2 with increasing equity is not associated

with any additional costs. In particular, the household would ask for a higher

risk premium if the bank granted more funds as loan to Firm 2 according to the

constraints determining Case 2.

The parameter values considered in Table 3.2 are those of the base case, as given in

Table 4.1 except for the values of WB. The threshold for WB according to (3.3.9)


Table 3.2: Optimal choices given Case 4 vs. equilibria

The upper panel reports the optimal choices given Case 4 for different levels of bank capital WB. Thethreshold for WB according to (3.3.9) amounts to 2040.14 The choices given Case 4 are contrastedwith the actual optimal choices by the bank, i.e. the equilibria, shown in the lower panel. Exceptfor WB, parameter values are those of the base case, as given in Table 4.1.

Panel A: Case 4

WB `∗(4) R∗D(4) D∗(·, ·; 4) E[WB(·, ·; 4)] U(WH) Eq./Res.

0 [0, 1] 1.0521 0 0 3150 Res. 9100 0.510638 1.05434 125.717 130.826 3150.14 (3.3.10)

1000 0.510638 1.07437 1205.49 1278.3 3163.39 (3.3.10)2040.14 0.510638 1.09735 2348.54 2543.68 3203.03 (3.3.10),

(3.3.11)4000 0.352999 1.09735 2348.54 4872 3203.03 (3.3.11)

10000 0.181481 1.09735 2348.54 12000 3203.03 (3.3.11)

Panel B: Equilibrium

WB `∗ R∗D j∗ D∗ E[W ∗

B ] U(W ∗H) Eq./Res.

0 0.585096 1.10905 1 1276.13 77.7021 3182.87 n/a100 0.571138 1.10802 1 1303.59 197.22 3185.06 n/a

1000 0.574687 1.08839 3 1545.95 1282.5 3180.5 n/a2040.14 0.510638 1.09735 4 2348.54 2543.68 3203.03 (3.3.10),

(3.3.11)4000 0.352999 1.09735 4 2348.54 4872 3203.03 (3.3.11)

10000 0.181481 1.09735 4 2348.54 12000 3203.03 (3.3.11)

amounts to 2040.14.

We note that it is by incidence that Case 4 prevails in equilibrium at this threshold.

For example, Case 2 arises in equilibrium in the scenario considered in Table 3.1,

although the bank’s equity WB clearly exceeds the threshold (3.3.9) which equals

58.7378 in that parametrization. Instead, increasing WB rather supports Case 2

as equilibrium, since Condition (3.3.2) guarantees that Condition (3.3.3) to obtain

Result 8 is negatively affected by WB.

Though the deposit-supply function Ds(`, RD; 4) is independent of WB in Case

4 according to Result 1, the household will not be generally indifferent between

different levels of WB if Case 4 occurs in equilibrium: if and only if WB is below the


threshold (3.3.9), the household appreciates increasing WB, i.e. the buffer function

that WB fulfills if the bank defaults. This property can be readily obtained from

Result 9:

Result 10. Assume Case 4 to be optimal. Then the optimal deposit volume strictly

increases in the bank’s initial equity WB, i.e.

dD∗

dWB

> 0

if Condition (3.3.9) holds and is independent of WB otherwise.

The proof is given in Appendix A.2.1.6. This result shows that there can be a

positive relation between the bank’s initial equity, WB, and the optimal deposit

volume in equilibrium even if the out-of-equilibrium deposit supply does not depend

on WB.

Considering once again equal success probabilities p1 = p2 = p and equal

productivity rates α1 = α2 = α, Case 4 changes its nature: the bank may also

choose ` = 12≡ α2

α1+α2as maximum given Case 4:


and that p1 = p2 = p and α1 = α2 = α hold. If

0 < WB ≤(pα−Rf ) · [(2p+ q)Rf − q4pα]

4p2α2(1− q4)γRf

(3.3.12)

holds, the maximum (`∗, R∗D, 4) given Case 4 is unique and characterized by the

solution`∗ = 1

2

R∗D =q4α+2Rf+

√[2Rf−q4α]

2+4q4(1−q4)α2γWBRf

2q4[2−(1−q4)αγWB ]

with

R∗D ≥ 1q4Rf

Remark 1= RD(4) ,

(3.3.13)

resulting in the following signs for the sensitivities of the deposit interest rate,

∂R∗D∂WB

> 0,∂R∗D∂γ

> 0,∂R∗D∂Rf

> 0,∂R∗D∂q4

< 0 . (3.3.14)


If Condition (3.3.12) is violated, αL∗1 6= αL∗2 may prevail, implying

`∗ ∈[

2pRf (pα−Rf )

q4(p2α2−R2f )+4p2α2(1−q4)γWBRf

,(pα−Rf )[q4pα−qRf ]+4p2α2(1−q4)γWBRf

q4(p2α2−R2f )+4p2α2(1−q4)γWBRf

]

with

`∗ ∈ (0, 1)

R∗D =2pαRf

q4(pα+Rf )> 1

q4Rf

Remark 1= RD(4) ,

D∗ =q4(p2α2−R2

f )

4p2α2(1−q4)γRf,

(3.3.15)

resulting in the following signs for the sensitivities of the deposit interest rate,

∂R∗D∂WB

= 0,∂R∗D∂γ

= 0,∂R∗D∂Rf

> 0,∂R∗D∂q

> 0,∂R∗D∂p

< 0,∂R∗D∂α

> 0 , (3.3.16)

and in the following signs for the sensitivities of the deposit volume,

∂D∗

∂WB

= 0,∂D∗

∂γ< 0,

∂D∗

∂q< 0,

∂D∗

∂p> 0,

∂D∗

∂α> 0 , (3.3.17)

For WB = 0, Case 4 can never arise in equilibrium, cf. Result 9. The solution

characterized by (3.3.13) and (3.3.15) is continuous in WB.

The derivation of the main formulæ are shown in Appendix A.2.1.7.

Unlike in the case of two different firms’ projects with respect to their risk-return

characteristics, two distinct, but otherwise equal projects let the bank either choose

the tuple (`, RD) such that both Case constraints are binding or such that none of

the constraints are binding. Thus, if none of the Case constraints are binding, the

bank can offer the interest rate on deposits that maximizes its expected final wealth

as in the unconstrained problem. To make this choice feasible, the bank is required

to have a sufficiently high amount of initial equity WB, characterized by (3.3.12).14

Since the return and the risks of the deposits are not affected as long as the bank’s

choice does not infringe the two Case constraints, the bank can freely allocate its

funds to both firms within Case 4, as given by (3.3.15).

This rationale, however, does not apply under Case 4 as soon as the firms’ projects

are different. Then the risk-neutral bank always grants as much funds as possible as

14The positive sign of the threshold (3.3.12) can be easily seen by considering the followingsequence of lower bounds, (2p + q)Rf − q4pα ≤ (1 + q)Rf − q4pα = (1 + q)Rf − (2p − q)pα >(1 + q)Rf − pα > 1 + q − 2p > 0, where Properties (3.2.5), (3.2.41), (3.2.8), and (3.2.6) as well as(3.2.7) are sequentially applied.


loan to Firm 2 if the expected return on the second firm’s loan exceeds that on the

first firm’s. Hence, the bank’s choice is always effectively restricted by the constraint

α1` [Du(RD; 4) +WB] ≥ Du(RD; 4)RD irrespective of its initial endowment with

WB.

This structural difference can also be read off by the formulæ in Result 11 compared

to those in Result 9 as follows: all formulæ shown by Result 11 can be directly

derived from those in Result 9 except for the upper bound on the optimal loan-

allocation rates in (3.3.15).

If the bank is constrained by its initial wealthWB according to the threshold (3.3.12),

both Case constraints are effectively binding, as it is the case under the general

parametrization analyzed in Result 9. Then the shadow costs of each binding

constraint reflects the gain of financial intermediation. That is, the shadow costs

are strictly positive if and only if

pα− q4R∗D

q4R∗D −Rf

(3.2.41)≡ pα− (2p− q)R∗D(2p− q)R∗D −Rf

>pα

Rf

(3.3.18)

holds. The left-hand side of the inequality shows the expected marginal gross profit

on deposits for the bank over the spread between the expected equilibrium return on

deposits q4R∗D and the risk-free interest rate Rf , which is the yield of the household’s

alternative investment. The right-hand side shows the bank’s expected gross return

on its loan portfolio over the risk-free rate. Thus, the ratio of (expected) gross

returns on the inequality’s right-hand side represents the ratio of (expected) gross

returns in autarky whereas the ratio on the inequality’s left-hand side embodies the

ratio of expected excess returns if the bank and the household bargain for deposits

under the conditions of Case 4.

The sensitivity of the optimal deposit interest rate RSD according to (3.3.13) with

respect to q4 results in

∂R∗D∂p

< 0, and∂R∗D∂q

> 0,

as q4 = 2p − q. These two sensitivities can be explained as follows: the deposit

interest rate strictly decreases in p as a higher success probability p means reduced

risks, i.e. a higher probability of full redemption of the deposits. In contrast,

increasing q is associated with an increasing correlation among both projects,

cf. (3.2.12). As under Case 4 deposits can be fully paid back as long as a single

loan is fully redeemed, a growing correlation among both projects endangers full


deposit redemption. However, the total effect of q4 on the total loan/deposit volume

is ambiguous as the direct impact is positive if RD > RD(4),

∂Ds(RD; 4)

∂q4

> 0 ,

thus (partially) off-setting the indirect effect via R∗D.

In contrast, if the Equilibrium (3.3.15) prevails, the total effect on the total

loan/deposit volume can be directly calculated: more risk in terms of q let the

optimal deposit volume shrink. Furthermore, with a higher success probability p,

the bank can collect more funds from the risk-averse household. Calculating the

effect of α on D∗ yields

∂D∗

∂α=

q4Rf

2p2(1− q4)α3γ> 0 .

Finally, the sensitivity with respect to p is given by

∂D∗

∂p=

p(p2α2 −R2f ) + (2p− q)(1− 2p+ q)R2

f

2p3α2(1− q4)2γRf

> 0 .

Concerning the Solution (3.3.13), the sensitivities with respect to WB, γ, and Rf can

be directly read off the Formula (3.3.13): the numerator strictly increases in these

variables, while the denominator strictly decreases in WB and γ and is independent

of Rf .

The sensitivities of the total loan/deposit volume are given by Result 10. Hence, if

and only if Condition (3.3.12) holds, the optimal deposit volume strictly increases

in WB, otherwise it remains independent of WB. Thus, the sensitivity of the total

loan volume under Case 4 can be summarized by

∆L(`∗, R∗D, 4)

∆WB

≥ 1 (3.3.19)

where the threshold given by (3.3.9) and (3.3.12), respectively, induces a kink at

which the slope of L(`∗, R∗D, 4) migrates from numbers strictly greater than one to

one.


3.3.1.5 Final Remarks

It can be shown that a positive, partial cushion effect is present in equilibrium

independent of the Case that may prevail. That is, a marginal increase translates

in equilibrium into a marginal, partial increase of the deposit volume:


and that p1 = p2 = p and α1 = α2 = α hold. Then a marginal change in bank capital

translates positively into a marginal change in deposits, i.e.

∂Ds(`∗, R∗D; j∗)

∂WB

≥ 0

where equality holds if and only if either `∗ = 0, or `∗ = 1, or Case 4 with WB

exceeding the thresholds given by (3.3.9) and by (3.3.12), respectively, prevail in

equilibrium.

Note that we have not presumed any specific Case to arise as the optimal one in

Result 12. Again, for given deposit interest rates and given deposit volumes, corner

solutions with either `∗ = 0 or `∗ = 1 would be optimal. Concerning Case 1, Result

12 is weaker than Result 7 as the former does a statement with respect to the partial

effect of the bank’s initial equity while the latter does to the total effect. So Result

12 can dispense with any additional assumptions (on αi).

Even if the project returns have equal distributions, one fails to generally

characterize the optimal solutions in Case 2 and 3, respectively. Likewise, it remains

opaque under which conditions a specific Case turns out to be the optimal one.

Specifically, the equilibrium may not be unique: first, Case 2 and 3 always lead to

the same expected final wealths of both the bank and the household so that agents

are always indifferent between Case 2 and 3. That is, Cases 2 and 3 become identical

except for the (optimal) loan-allocation rate. Table 3.3 provides an example where

the optimal choice in Case 2 can be characterized according to Result 8.

Second, multiple loan allocations may even be feasible within Case 4, namely if WB

exceeds the threshold (3.3.12). Third, the bank is indifferent between allocating its

funds between all four Cases if the firms’ projects are perfectly correlated, i.e. if

p = q holds.

Finally, we note that a change in the optimal Case j∗ may result in a discontinuity

in the optimal deposit volume Ds(`∗, R∗D; j∗) because a change in the Case means

that the bank picks out another deposit-supply function Ds(`, RD; j) to maximize


Table 3.3: Example of equal projects in the Bernoulli model w/o regulation

The upper panel shows an example set of parameter values where the success probabilities andthe productivity parameters of both projects are equal. The lower panel reports the bank’s optimalchoices in each Case j = 1, . . . , 4. The equilibrium is characterized by Cases 2 and 3, respectively.


p q α WB γ WH Rf

0.92 0.9 1.15 10 0.001 3000 1.05


`∗(j) R∗D(j) j D∗(·, ·; j) E[WB(·, ·; j)] U(WH) Eq./Res.

12 1.14547 (Case) 1 48.4323 10.5472 3150.33 Res. 60 1.14564 (Case) 2 41.2513 10.7456 3150.08 Res. 81 1.14564 (Case) 3 41.2513 10.7456 3150.08 n/a12 1.11782 (Case) 4 10.5929 10.6568 3150.00 (3.3.13)

its expected wealth associated with that Case. But, choosing another Case with its

associated deposit supply function formally corresponds to picking out another local

maximum, cf. (3.2.46). Thus, even marginal changes in the optimal loan-allocation

rate `∗ and in the deposit interest rate R∗D may result in a break in the optimal

deposit volume. As those marginal changes arise due to variations in exogenous

parameters, a shock, as discussed in Sections 2.1 and 2.2, may result in jumps in the

total loan/deposit volume. Examples are provided by Figures 4.1, 4.11, 4.19, and

4.20.

3.3.2 The Equilibrium with Regulation by Fixed Risk

Weights


In this paragraph, we analyze the bank’s choice under regulation schemes where

risk weights are assigned to each asset the bank holds on its banking book. This

approach has already been put forward under the Basel I Accord and has been

refined under the Basel II Accord, known as the Standardized Approach. The risk

weights are supposed to account for the credit risk each asset bears. In our model,


Table 3.4: Risk weights for corporates in the Standardized Approach

This table shows the risk weights for claims on corporates that have been set forth by BCBS (2004,Art. 66). The symbol cS represents the credit grade of the country of incorporation.

Credit grade AAA to AA- A+ to A- BBB+ to BB- lower than BB- UnratedRisk weight 0.2 0.5 1 1.5 max1, cS

we consider the loans as claims on corporates in the sense of Basel II. Thus we use

the risk weights ci proposed in Article 66 of the Basel II Accord (BCBS 2004). They

are presented in Table 3.4 where cS represents the risk weight of the country of

incorporation.

We do not consider any other risks. According to Article 40, banks are required to

maintain a capital ratio that must not fall below

c = 8% (3.3.20)

where that ratio is defined as regulatory capital, here simply WB, over risk weighted

assets, here given by c1L1 + c2L2, that is the following requirement

c · (c1 · L1 + c2 · L2) ≤ WB

must be met. The regulatory parameter c is also known as the Cooke ratio. This

regulatory constraint can be re-written as a constraint on the total deposit volume:

Ds(`, RD; j) ≤ k(`) ·WB ,

where k(`) =1− c · [c1`+ c2(1− `)]c · [c1`+ c2(1− `)] , (3.3.21)

with k(`) ∈ [22

3, 61.5] .

Due to (3.2.5), Loan 2 has a higher probability of default and a higher volatility on

returns than Loan 1. Thus,

c1 ≤ c2 (3.3.22)

if we assume credit ratings decrease in default probabilities. If both loans are

equally weighted, the choice of the loan-allocation rate ` does not affect the allowed

maximum volume of deposits and loans. If both loans are weighted differently,

c1 < c2, a shift in the loan allocation in favor of the less risky loan, namely Loan 1,


causes the deposit-to-equity-ratio k(`) to increase by

k′(`) =c2 − c1

c · [c1`+ c2(1− `)]2

with k′(`) ∈ (5

3, 406.25] , if c2 > c1 , (3.3.23)

having presumed the risk weights to take values according to Table 3.4. The higher

the loan-allocation rate `, the higher is the effect, i.e. the marginal benefit from

increasing the loan-allocation rate ` increases:

k′′(`) =2(c2 − c1)2

c [c1`+ c2(1− `)]3> 0 , if c1 6= c2 . (3.3.24)

Thus, the bank exhibits increasing returns to scale in terms of a higher business

volume in exchange for choosing less risky portfolio compositions. The costs are a

loss of expected return on its portfolio.

The bank’s decision problem under regulation by (exogenous) risk weighting of assets

can be formulated as follows (Buhler/Koziol/Sygusch, 2008, p. 147, Maximization

Problem (19)):

max`, RD

E[WB(`, RD; j)

]

s.t. ` ∈ [0, 1] (3.3.25)

∃j : (Ds(`, RD; j), `, RD) ∈ Cj

Ds(`, RD; j) ≤ k(`) ·WB .

We will refer to (`S, RSD; jS) as the equilibrium, or the bank’s optimal choice under

the Standardized Approach. Regulation is said to be binding if the regulatory

constraint holds with equality, i.e. Ds(`, RD; j) = k(`) ·WB.

Result 13. An equilibrium (`S, RSD; jS) always exists.15

Clearly, the existence result as shown by Result 5 carries over to the case of

regulation: the loan-allocation rate ` is restricted to [0, 1], the maximum interest rate

on deposits RD is bounded from above by Result 2, and the definition of each Case j

is based on a compact set Cj in `-RD space given a specified deposit-supply function.

Additionally, the regulatory constraint imposes an additional definite bound on the

set of feasible `-RD combinations.

15Buhler/Koziol/Sygusch (2008, p. 147).


If the household is not constrained by its initial wealth WH and if a local maximum

(`S, RSD; jS) to Problem (13) exists, the maximum (`S, RS

D; jS) satisfies the Kuhn-

Tucker conditions and a multiplier λ ≥ 0 exists such that16

∂E[WB(`S, RSD; jS)]

∂`= λ ·

[∂Du(`S, RS

D; jS)

∂`− k′(`S) ·WB

]

∂E[WB(`S, RSD; jS)]

∂RD

= λ · ∂Du(`S, RS

D; jS)

∂RD

λ ·[Ds(`S, RS

D; jS) − k(`S) ·WB

]= 0

holds. For this exposition, we have assumed that neither `S hits the borders of the

unit interval nor (`S, RSD; jS) lies at the boundary of CjS ,

If λS > 0 holds, regulation is binding and λS > 0 is the shadow cost of forgoing

profit opportunities by reducing the business volume to[k(`S) + 1

]·WB.

To assess the potential pro-cyclical impact regulation has on lending, we consider

the sensitivities of the equilibrium deposit (loan) amount to exogenous parameters.

Changes in those parameters are considered as shocks. Such a shock is a loss (or

gain) in initial equity by realized credit losses (or by loan redemptions exclusively as

promised) in a notional preceding period. If regulation is binding and if the optimal

Case jS does not vary, the effect of a change in initial equity on the deposit volume

is given bydDs(`S, RS

D; jS)

dWB

= k(`S) + k′(`S) · d`S

dWB

·WB . (3.3.26)

One can identify two sources affecting this sensitivity. The first or direct effect

equals the bank’s deposit-to-equity ratio, reflecting the direct marginal impact of

initial equity on the optimal deposit volume. The second, or indirect effect, is due to

a modification in the loan-allocation rate `S. This effect becomes more pronounced

as the bank becomes more capitalized and it may affect the sensitivity positively or

negatively. If it is optimal for the bank to grant only a single loan, d`S

dθ= 0 prevails

and the latter effect vanishes. It also vanishes if loans are equally weighted, since

k′(`S) = 0 for c1 = c2 holds. Then, comparable to Basel I, the bank’s optimal choice

of the loan-allocation rate does not affect the feasible maximum loan and deposit

volume and the sensitivity of lending to equity-shocks is given by

dLS

dWB

=1

cc1

∈ [25

3, 62.5] .

16For an exposition we refer to Mas-Collel/Winston/Green, (1995, p. 959, Theorem M.K.2), orTakayama (1994, p. 93f).


If we consider shocks with respect to firms’ productivity, to firms’ probabilities of

success, or to the common success probability q, the direct effect is canceled out

such that the complete sensitivity is given by

dDs(`S, RSD; jS)

dθ= k′(`S) · d`

S

dθ·WB (3.3.27)

where θ is one of the parameter pi, q, or αi, i = 1, 2. The same reasoning applies to

shocks that simultaneously affect several of these parameters. The numerical studies

concerning these parameters in the next chapter, cf. Sections 4.3.1 and 4.4.1, are

based on such common multipliers. The deposit (loan) volume is only affected by

the second order effect. The total loan and deposit volumes remain unaffected if

loans are equally weighted, due to k′(`S) = 0, or as long as it is optimal for the bank

to grant only a single loan, implying d`S

dθ= 0.

To conclude, these second-order effects imposed by a risk-sensitive regulation

enhance the sensitivity of total lending toward shocks compared to regulation with

fixed and equal risk weights. Figures 4.1 in connection with 4.2, 4.11, and 4.19

provide examples. On the level of single loans, the opposite may be true, as shown

by Figure 4.23 because, under fixed risk weights, loan-allocation rates move while

the total loan volume remains fixed.

But risk-sensitive regulation need not necessarily add to pro-cyclicality in the sense

that lending volumes react more strongly on shocks than lending volumes granted by

an unregulated bank as the examples above show. The same applies to a variation

of this model with loan-portfolio returns approximated by the normal distribution.

The associated numerical results will be discussed in Section 6.6.

To grasp some further insights into the impact regulation has on the bank’s optimal

choices, we are going to outline some equilibria that can be stated in closed form.

In general, there is no explicit characterization possible for Cases 1 to 3. For Case 1

we consider again the case of two equally distributed projects, for Case 2 the corner

solution in `S = 0.


Such as in Paragraph 3.3.1.2, p. 67, we analyze the equilibrium under Case 1 if both

firms’ projects are equal concerning their expected return and their variance:

Result 14. Let p1 = p2 = p, α1 = α2 = α, and c1 = c2. Suppose that regulation

is binding, the household is not constrained by its initial wealth and that Case 1 is


optimal. Then the bank allocates its funds equally as loans to both firms, i.e. `S = 12,

and promises

RSD =

cc1q + (2− cc1)(p− q)qαγWB −√QS

1 (c1, α)

2(1− cc1)(1− q)qγWB

as gross interest rate on deposits to the household where

QS1 (c1, α) = c2c2

1q2 [1 + (p− q)αγWB]2

+ 4cc1(1− cc1)q [(p− q)α− (1− q)Rf ] γWB

− 2(1− cc1)q(p− q)(1− 2p+ q)α2γ2W 2B .

The total loan volume LS is given by LS = 1cc1·WB. The choice (`S, RS

D) is unique

given Case 1. The signs of the sensitivities of the deposit interest rate RSD are given

as follows,

∂RSD

∂WB

> 0,∂RS

D

∂γ> 0,

∂RSD

∂Rf

> 0,∂RS

D

∂c< 0,

∂RSD

∂c1

≡ ∂RSD

∂c2

< 0 . (3.3.28)

The sensitivities of the optimal deposit interest rate RSD can be explained as follows:

increasing equity WB results in a higher interest rate, despite of the risk-reducing

effect of the bank equity’s buffer function, because RD is the only variable left to

increase the deposit volume with a growing regulatory constraint k(`S) ·WB as the

optimal loan-allocation rate `S does not change in equilibrium. Likewise, a higher

risk weight ci or a higher Cooke ratio c reduces RSD because the bank can only reduce

its total loan volume, LS = 1cc1·WB, by lowering the deposit supply via the interest

rate promised.

RSD increases with Rf as the household can substitute bank deposits for sovereign

bonds. Were RSD too low relative to Rf , the bank might not be able to fully exploit

the regulatory constraint.

The derivation of the result is shown in Appendix A.2.2.1.

Finally, we can draw the following comparison between regulation and non-

regulation if Case 1 is optimal for equally distributed firm projects:

Result 15. Suppose that the assumptions of Results 6 and 14 hold. Then regulation

makes the total loan volume, LS, and the deposit interest rate RSD pro-cyclical

concerning changes in c and ci. Except for WB, the regulated total loan volume is

not pro-cyclical concerning shocks in any other parameter. Furthermore, regulation

results in a smaller interest rate on deposits, RSD < R∗D. Assume additionally


α ≤ 43. Then regulation also makes the total loan volume, LS, on average pro-

cyclical concerning shocks in WB.

Pro-cyclicality concerning the regulatory parameters c and c1 is trivial. Furthermore,

it is obvious from LS = 1cc1·WB that reactions of the total loan volume to shocks

other than shocks in WB, c, and ci are dampened by regulation. The optimal total

loan volume does not depend on the optimal loan-allocation rate because both risk

weights are equal, c1 = c2. Consequently, the total loan volume does not depend

on any further parameters via the loan-allocation rate. Note that `S = 12

holds also

under regulation due to c1 = c2.

Shocks in the bank’s initial equity WB are pronounced by regulation and affect the

total loan volume more heavily. This mechanism is mainly due to the assumption

that the bank may not hold any capital buffers in excess of the regulatorily necessary

capital. Thus, binding regulation always implies that the total optimal loan volume

goes along the upper bound[k(`S) + 1

]·WB which is fixed in the case considered

by Result 15 to 1cc1WB. But there will be some equity threshold WB beyond which

regulation is no longer binding. Likewise, the regulation prevents the bank from

doing intermediation when WB = 0. Thus, the total loan volume evolves linearly

and strictly increases in WB for all WB ∈ [0,WB]. At WB = WB, regulation ceases

to bind and thus from WB = WB on, the total loan volume coincides with the

unregulated one. Now consider the unregulated bank in Case 1. It can collect

deposits and grant loans even if it does not have any equity, i.e.

Du(1

2, R∗D; 1)

∣∣∣∣WB=0

=(p− q)α + qR∗D −Rf

γσ21

> 0 .

In WB = WB, the unregulated bank grants a total loan volume as high as 1cc1·WB

by definition. In between, L∗ > LS holds, and L∗ is strictly increasing if α ≤ 43

is

assumed (cf. Result 7 and the associated Appendix A.2.1.3). Hence, the positive

slope of LS in WB exceeds on [0,WB] on average the positive slope of L∗, i.e.

1

c · c1

=LS(WB)− LS(WB)

WB −WB

>L∗(WB)− L∗(WB)

WB −WB

Result 7> 1, (3.3.29)

for all WB ∈ [0,WB). Thus, regulation affects the total loan volume on average

pro-cyclically according to Definition (2.2.4).

The effect on the deposit interest rate, RSD < R∗D, has already been shown in

Appendix A.2.2.1. It is simply due to the fact that the deposit interest rate RD is

the only control variable left to the bank to meet the regulatory constraint 1−cc1cc1·WB


as it is optimal for the bank to choose ` = 12

with or without a binding regulatory

constraint. Furthermore, this result clearly reflects the monopolistic power the bank

is assumed to have on the deposit market.

Note that if both firms’ projects can be distinguished according to their risk and

return, the loan-allocation rate without regulation `∗ and the loan-allocation rate

under a binding regulatory constraint `S will differ from each other. Hence, in

general, it remains an open question of how regulation will affect the risk-neutral

bank owners’ trade-off between the loan-allocation rate and the deposit interest rate.

The numerical analyses in Chapter 4 suggest that regulation affects both the deposit

interest rate and the loan-allocation rate negatively from the household’s point of

view, i.e. RSD < R∗D and `S < `∗ prevail.

3.3.2.3 Characterization if Case 2 with `S = 0 Prevails

One such equilibria could be that the bank optimally grants all its funds as loans to

the riskier firm:


and that the expected final wealth of the bank, E[WB(`, RD; j)], is maximized in

`S = 0 on R and that regulation is binding. Then

RSD =

cc2p2 −√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

2(1− cc2)p2(1− p2)γWB

>1

p2

Rf

is the optimal interest rate on deposits resulting in the sensitivities

∂RSD

∂WB

> 0,∂RS

D

∂γ> 0,

∂RSD

∂Rf

> 0,∂RS

D

∂p2

< 0,∂RS

D

∂c< 0,

∂RSD

∂c2

< 0 . (3.3.30)

The total loan volume LS is given by LS = 1cc2WB.

The sensitivity of the optimal deposit interest rate RSD with respect to p2 can be

explained as follows: an increasing success probability p2 means reduced risks if

anything else is held fixed. Thus, an increasing success probability p2 results in

a lower interest rate. With respect to the remaining sensitivities, we refer to the

discussion following Result 14.

The total loan volume LS strictly increases in WB, and strictly decreases in the

regulatory parameters c and c2. Under the assumption that there are parameter

values such that the unregulated as well as the regulated bank chooses ` = 0 as


optimal loan-allocation rate one immediately obtains from the Results 8 and 16:

Result 17. Assume that the conditions of Result 8 and 16 hold. Then regulation

makes the total loan volume, LS, pro-cyclical concerning shocks in WB, and

concerning changes in c and c2. But the regulated total loan volume is not pro-

cyclical concerning shocks in any other parameter. Regulation makes the deposit

interest rate pro-cyclical concerning shocks in WB, γ, Rf , c, and c2. The regulated

deposit interest rate is not pro-cyclical concerning shocks in any other parameter.

Concerning the total loan volume, this result is trivial. The same applies to the

comparison of the sensitivities of the deposit interest rates with respect to the bank’s

initial equity WB, the gross return α2, and the household’s risk-aversion parameter

γ. Concerning the sensitivity to Rf we refer to the Appendix A.2.2.3.


Next, we examine the characteristics of the equilibrium if it is optimal for the bank

to structure its balance sheet such that either promised loan redemption suffices to

fully pay-off the depositor (Case 4). First, we provide bounds on the productivity

parameter (equilibrium loan interest rate) αi beyond which regulation effectively

becomes binding.

Result 18. Let WB > 0. If

α2 >2 ·

1− c ·[c1`

S + c2(1− `S)]

q4

·Rf ≥1.76

q4

·Rf (3.3.31)

holds true, the Regulatory Constraint (3.3.26) becomes binding in Case 4.

If gross returns on loans, αi, are too low, regulation does not become binding in Case

4. Rather, the bank’s optimal choice is limited by the Case constraints such that the

highest feasible deposit volume under regulation is not met. Increasing αi implies a

higher leeway on the bank’s choice opportunities, in particular a higher leeway on

the deposit volume. Consequently, if αi is sufficiently high, optimal choices will hit

the Regulatory Constraint (3.3.26).

This mechanism may prevent the bank from choosing Case 4 under regulation if

αi lies below (3.3.31). Since other Cases will allow for higher deposit volumes,

specifically for DS(`, RD; j) = k(`) ·WB > DS(4), j = 1, 2, 3, it is likely that the

bank can determine the loan-allocation rate ` and the deposit interest rate RD such


that it overall profits from granting more loans and issuing more deposits than it is

feasible in Case 4.

The formal exposition of the argument can be found in Appendix A.2.2.4. Table

3.5 illustrates of how the optimal choice in Case 4 under regulation and of how the

equilibrium outcome under regulation are affected by αi. To present this effect more

pronounced, an equal parametrization for both projects is considered. αi = 1.95301,

i = 1, 2, is just that value for αi at which both the Case constraints as well as the

regulatory constraint hold with equality. Thus, this point marks the transition

between the regime with both Case constraints only binding to that one in which

the regulatory constraint is exclusively binding.

The following result characterizes the equilibrium under Case 4 if regulation is

binding:

Result 19. Suppose Case 4 be optimal, that the household is not constrained by its

initial wealth, that the optimal loan-allocation rate is from the interior of the unit

interval, and that regulation is binding. If

p2α2

p1α1

≥ c2

c1

> 1 , wherep2α2

p1α1

(3.2.5)> 1 , (3.3.32)

holds, the equilibrium is characterized by the following loan and deposit redemption

volumes

α1LS1 = DSRS

D ∧ α2LS2 > DSRS

D , (3.3.33)

where

`S =

c2c2−c1 · (q4R

SD −Rf )− (1−cc2)

c(c2−c1)· q4(1− q4)γWB(RS

D)2

q4RSD −Rf + q4(1− q4)γWB(RS

D)2,

RSD =

cc2q4α1 + c(c2 − c1)Rf −√QS

4 (c1, c2, α1)

2 [c(c2 − c1)q4 + (1− cc2)q4(1− q4)α1γWB], (3.3.34)

with

QS4 (c1, c2, α1) = c2 [c2q4α1 − (c2 − c1)Rf ]

2 − 4cc2(1− cc2)(1− q4)q4α21γWBRf .

Ifp2α2

p1α1

≥ c2

c1

= 1 , (3.3.35)


Table 3.5: Example illustrating Result 18

The upper panel reports parameter values used in this example. The intermediate panel reports thebank’s optimal choices in Case 4 for different productivity parameter values under regulation, thelower panel shows the equilibrium outcomes.


p1 = p2 q WB γ WH Rf c c1 = c2

0.992 0.985 100 0.008 3000 1.05 0.08 1

Panel B: Optimal solutions given Case 4

α1 = α2 `S(4) RSD(4) DS(·, ·; 4) E[WSB(·, ·; 4)] U(WH) Eq./Res.

1.1 0.5 1.05202 109.557 113.528 3150.05 (3.3.13)1.76Rf

2p−q = 0.5 1.0573 698.717 727.672 3152.18 (3.3.13)= 1.84985

2(1−c·c1)Rf

2p−q = 0.5 1.06036 1035.33 1081.36 3154.82 (3.3.13)= 1.933931.953005 0.5 1.06142 1150 1202.32 3155.95 (3.3.13)

2(1−c·0.5)Rf

2p−q = [ 0.483892, 1.06142 1150 1282.93 3155.95 Res. 21= 2.01802 0.516108 ]

2(1−c·0.2)Rf

2p−q = [ 0.47209, 1.06142 1150 1345.49 3155.95 Res. 21= 2.06847 0.52791 ]

Panel C: Equilibria

α1 = α2 `S RSD jS DS E[WSB ] U(WS

H) Eq./Res.1.1 0.5 1.07669 1 538.837 120.723 3156.38 Res. 6

1.76Rf

2p−q = 0.4707 1.06427 2 1150 1072.08 3157.75 n/a= 1.84985 0.5293 3

2(1−c·c1)Rf

2p−q = 0.49464 1.06232 2 1150 1177.82 3156.78 n/a= 1.933931.953005 0.5 1.06142 4 1150 1202.32 3155.95 (3.3.13)

2(1−c·0.5)Rf

2p−q = [ 0.483892, 1.06142 4 1150 1282.93 3155.95 Res. 21= 2.01802 0.516108 ]

2(1−c·0.2)Rf

2p−q = [ 0.47209, 1.06142 4 1150 1345.49 3155.95 Res. 21= 2.06847 0.52791 ]


holds, the equilibrium satisfies (3.3.33), where

`S =cc1q4α1 −

√QS

4 (c1, c1, α1)

2(1− q4)q4α21γWB

, (3.3.36)

with

QS4 (c1, c1, α1) = c2c2

1q24α

21 − 4cc1(1− cc1)q4(1− q4)α2

1γWBRf .

The deposit interest rate can be directly derived from (3.3.34).

If and only if

1(3.2.5)> p1α1

p2α2≥ c1

c2+ c2−c1

c2· Rfp1α1· c(c2−c1)Rf−cc1q4α2+

√QS4 (c2,c1,α2)

3cc1(c2−c1)Rf+cc1q4α2−4(1−cc1)(1−q4)α2γWBRf−√QS4 (c2,c1,α2)

,

(3.3.37)

where

QS4 (c2, c1, α2) = c2 [c1q4α2 + (c2 − c1)Rf ]

2 − 4cc1(1− cc1)(1− q4)q4α22γWBRf

holds, the equilibrium is characterized by the following loan and deposit redemption

volumes

α1LS1 > DSRS

D and α2LS2 = DSRS

D (3.3.38)

where

`S =(α2 −RS

D)(q4RSD −Rf ) + q4(1− q4)α2

(RSD

)2γWB[

(q4RSD −Rf ) + q4(1− q4) (RS

D)2γWB

]α2

,

RSD =

c(c2 − c1)Rf − cc1q4α2 +√QS

4 (c2, c1, α2)

2q4 [c(c2 − c1)− (1− cc1)(1− q4)α2γWB]. (3.3.39)

The asymmetric allocation of funds to both loans according to (3.3.33) and (3.3.38),

respectively, is due to regulation. Without regulation, it may happen that both

Inequalities (3.2.19) that define the feasibility set C4 hold with equality, and (3.3.10)

prevails. This is not the case when regulation is supposed to be binding, unless at

that point where the Regulatory Constraint (3.3.21) just starts to hold with equality.

Else, either the tuple (`∗, R∗D; 4) according to (3.3.10) induces the deposit volume

violating the Regulatory Bound (3.3.21) and thus renders the solution unfeasible

under regulation, or the bank does not fully exploit the total loan/deposit volume

that is feasible under regulation, contradicting the assumption of binding regulation.

Table 3.5, Panel B, provides an example with respect to different values of the

productivity parameter and two equally distributed projects. At α = 1.953005,


both the regulatory constraint and the Case constraints hold with equality. Below,

only the Case constraints are fully exploited, above only the regulatory constraint.

Under binding regulation, the optimal loan-allocation rate `S(4) is mainly driven by

the relation of both risk weights c1 and c2 to the expected gross returns on both loans,

p1α1 and p2α2. As deposit supply does not depend on the loan-allocation rate under

Case 4, the Regulatory Constraint (3.3.21) is only affected by the loan-allocation

rate via the risk weighting function c1`+ c2(1− `). Thus, the risk-neutral bank has

only an incentive to grant a higher loan volume to Firm 1 than to Firm 2, if there

is a leeway on Constraint (3.3.21), i.e. c1 < c2, that is big enough to compensate for

granting more to the less profitable firm. In this model, a compensation in favor of

the bank is a lower deposit interest rate to be charged. As the deposit interest rate

depends on the risks associated with deposits and on the household’s risk aversion,

any condition that results in α1LS1 > α2L

S2 should depend on these parameters.

Condition (3.3.37) does. It is necessary and sufficient to obtain α1LS1 > α2L

S2 .

In contrast, Condition (3.3.32) does neither account for the household’s risk aversion

nor for the credit risk of loans and deposits. It results in a loan-allocation rate and

in a deposit interest rate such that α1LS1 < α2L

S2 prevails. Hence, a condition solely

depending on the risk weights ci and on the expected returns piαi is too weak to

sustain a shift in the portfolio composition in favor of the less risky, but also less

profitable Loan 1.

Details to the derivation of Result 19 can be found in Appendix A.2.2.5. Note

furthermore, that by the sufficiency of Condition (3.3.32), Solution (3.3.34) can also

prevail as equilibrium for p2α2

p1α1< c2

c1.

Result 19 allows the following conclusion:

Result 20. Suppose as equilibria the choices characterized by Results 9 and 19.

Then regulation makes the total loan volume, LS, and the deposit interest rate RSD

pro-cyclical concerning changes in c and ci and regulation results in a smaller interest

rate on deposits, RSD < R∗D.

Suppose that either Assumption (3.3.32) or (3.3.35) holds. Then the regulated

deposit interest rate is not pro-cyclical concerning shocks in α2.

In contrast, if Assumption (3.3.37) holds, the regulated deposit interest rate is not

pro-cyclical concerning shocks in α1. Furthermore, the interest rate, and thus the

deposit volume, strictly increase in WB as long as regulation is binding,

dRSD

dWB

> 0, anddDS

dWB

> 0 .


As the deposit-supply function depends only on RD under Case 4, as the total

loan/deposit volume is assumed to be effectively constrained by regulation, i.e. DS <

D∗, and, finally, as deposit supply strictly increases in RD under both regimes in

equilibrium, the relation RSD < R∗D holds. The sign of the sensitivity

dRSDdWB

is derived

in Appendix A.2.2.6. The sign of the sensitivity of the total deposit volume directly

results from the sign of the deposit rate’s sensitivity. Concerning the fact that the

deposit-supply function increases in RD in equilibrium, we refer to Result 5 and

Result 10.

Finally, Case 4 could also prevail in equilibrium if both firms are equal. The

equilibrium can then be characterized as follows:


and that p1 = p2 = p, α1 = α2 = α, and c1 = c2 hold, and that regulation is binding.

Then the bank allocates its funds, given Case 4, within the range

`S ∈[cc1q4α−

√QS4 (c1,c1,α)

2q4(1−q4)α2γWB, 1 +

2cc1(1−cc1)Rf

cc1q4α−√QS4 (c1,c1,α)

− cc1(1−q4)αγWB

]

and promises

RSD =

cc1q4α−√QS

4 (c1, c1, α)

2(1− cc1)q4(1− q4)αγWB

(3.3.40)

as gross interest rate to the household, where

QS4 (c1, c1, α) = c2c2

1q24α

2 − 4(1− cc1)cc1q4(1− q4)α2γWBRf .

The total loan volume LS is given by LS = 1cc1·WB. The signs of the sensitivities

of the deposit interest rate RSD are given as follows,

∂RSD

∂WB

> 0,∂RS

D

∂γ> 0,

∂RSD

∂Rf

> 0,∂RS

D

∂q4

< 0,∂RS

D

∂c< 0,

∂RSD

∂c1

≡ ∂RSD

∂c2

< 0 .

(3.3.41)

The sensitivity of the optimal deposit interest rate RSD with respect to q4 is discussed

on p. 77 following Result 11. With respect to the remaining sensitivities, we refer

to the explanations following Result 14.

The equality of the projects’ returns and risk results in a range of feasible loan-

allocation rates. The lower bound for this range can also be directly derived from

(3.3.36) by setting p1 = p2 and α1 = α. The upper bound of this range stems

from the appropriate Case constraint. The optimal deposit interest rate can also be

directly traced back to Formulæ (3.3.34) and (3.3.39), respectively. Note that the


assumption of a binding regulatory constraint requires the productivity parameters

at least to exceed (3.3.31) as stated in Result 18.

The derivation of the solution is discussed in Appendix A.2.2.7.

Result 15, that refers to Case 1, applies as well, in particular RSD < R∗D holds as

outlined in Appendix A.2.2.7. The reference equilibrium is that characterized in

Result 11.

But the arguments concerning pro-cyclicality on average of the total loan/deposit

volumes with respect to shocks in WB must be slightly altered. First, the upper

bound on α assumed in Result 15 and derived in Result 7 can be dispensed with.

Instead, α must exceed (3.3.31) in order to make the regulatory constraint be

binding.

Second, we must consider that, without regulation, the optimal deposit volume also

goes to zero in Case 4 if the bank’s initial equity, WB, goes to zero. Hence, Case 4

can only occur from a given threshold WB > 0 on in the laissez-faire equilibrium.17

Third, there is the Bound (3.3.12) concerning WB, beyond which the unregulated

deposit volume is independent of WB, and below which the unregulated deposit

volume strictly increases in WB, as summarized by Formula (3.3.19) concerning

the total loan volume. Let this bound beWB. Fourth, the deposit volume strictly

increases inWB under binding regulation by a constant rate of 1−cc1cc1

. Finally, binding

regulation implies by definition DS < D∗. Let WB be the threshold at which

regulation ceases to bind.

Merging these observations leads to the following result:

There is a range of values of WB, WB ∈ [0, WB], where the optimal loan/deposit

volumes are strictly increasing with regulation. The slope of the unregulated deposit

volume D∗, which is not necessarily positive (cf. Results 7 and 12), is on average

lower than the slope of the regulated volume DS. This is due to the fact that

D∗ > DS = 0 holds in WB = 0 and that DS catches up to D∗ by a constant, positive

slope of 1−cc1cc1

as both loan volumes are equally weighted. Without regulation, we

will observe any of the Cases 1 to 3, but not Case 4.

If regulation is still binding for WB ∈ [WB,WB], the regulated total loan/deposit

volume further catches up to the unregulated volumes by its constant slope, whereas

D∗ is strictly increasing in WB as we have assumed Case 4 to hold in the laissez-faire

equilibrium. Hence, regulation is supposed to influence total lending in a pro-cyclical

manner on average. The same reasoning applies to WB <WB if WB ≤ WB < WB

17Cf. Result 9 and the subsequent discussion on p. 73.


holds and thus the unregulated total loan volume is unaffected by changes in WB.

If we further consider the different dependencies of RSD according to (3.3.40) and of

R∗D according to (3.3.13) and (3.3.15), respectively, concerning the same parameters,

the effects of regulation can be summarized as follows:

Result 22. Suppose that the assumptions of Results 11, 18, and 21 hold. Then

regulation makes the total loan volume, LS, and the deposit interest rate RSD pro-

cyclical concerning changes in c and ci. Furthermore, the regulated total loan

volume is pro-cyclical on average concerning shocks in WB, but LS is not pro-cyclical

concerning shocks in any other parameter. The regulated deposit interest rate is not

pro-cyclical concerning shocks in α. Above all, regulation results in a smaller interest

rate on deposits, RSD < R∗D. Let (3.3.15) be the equilibrium without regulation. Then

the regulated deposit interest rate RSD is pro-cyclical concerning changes in WB and

γ.

3.3.3 The Equilibrium with Regulation by a Value-at-Risk

Approach


As already pointed out in Section 1.3.1, the assignment of risk weights according to

the internal-ratings-based approach can be interpreted as the calculation of credit

VaR measures per loss given default and per exposure at default for each single

loan. Instead of applying the thus derived risk weights, we determine the VaR of

the whole portfolio based on the default probabilities and default correlations given

by this model.

Unexpected losses to the bank’s loan portfolio are defined as all realizations of the

loan portfolio’s value below the expected redemption E[L], i.e. unexpected losses

are given by

E[L] − L > 0 , (3.3.42)

where L = α1 · X1 · L1 + α2 · X2 · L2.

A VaR constraint requires that unexpected losses not exceed a given threshold by

a probability of p. This threshold is linked to the bank’s initial equity WB by a

multiplier τ > 0. Thus, the bank is required to meet

P(

E(L)− L ≥ τWB

)≤ p . (3.3.43)


The lower the τ , the more often unexpected losses exceed the threshold τWB for

given loan volumes. Hence, a lower τ requires the bank to decrease its loan volumes

if the level of confidence p is fixed. Thus, τ quantifies the strictness of regulation.

Furthermore, the value of τ can be regarded as the required resilience of equity

to unexpected losses. If τ = 1, the bank’s initial equity should fully absorb all

(un)expected losses up to a given probability p. With τ < 1, the initial equity

can withstand losses such that a strictly positive final equity value remains within

probability p. For τ > 1, total wipe-outs of the initial equity are tolerated within

the probability p.

Similar to the parameter τ , the strictness of the Regulatory Constraint (3.3.43)

increases with decreasing initial equity WB: the lower WB is, the larger the domain

of loan volumes becomes for which the inequality L ≤ E(L)−τWB continues to hold

and accordingly P(

E(L)− L ≥ τWB

)for given values of ` and RD rises. In order

to reach the confidence level p again, the domain of previously feasible loan/deposit-

volumes must shrink. Thus the initial equity WB has a pro-cyclical effect on the

total loan volume L, given fixed ` and RD.

The question is how the VaR Constraint (3.3.43) can be aligned with the internal-

ratings-based approach in Basel II.

The latter requires that fractional losses conditional on a realization of the macro-

factor not exceed a fixed threshold by 1−p = 0.999. The calculation of this threshold

loss rate is given in BCBS (2004, Art. 272).18 As this quantile is measured in terms

of loss rates it is multiplied by the exposure at default, yielding an absolute measure

for the VaR at the 0.1%-level. Subtracting expected losses results in unexpected

losses. More precisely, the IRB formula thus determines unexpected losses at the

0.1%-level that must be fully covered by the bank’s (regulatory) capital.19

Since unexpected losses are defined according to (3.3.42), τ = 1 is most suitable if

(3.3.43) is to reflect the IRB approach. In Chapter 4, the numerical studies make

use of τ = 4 to exaggerate the sensitivity of capital requirements, especially with

respect to WB. We will come back to τ = 1 in Chapters 6 and 7.

Depending on the ordering of the two loans and on the confidence level p, several

constraints arise from the credit VaR Condition (3.3.43). They are shown in detail

in Table 3.6. Their derivation can be found in Appendix A.2.3.1.

18Schonbucher (2000) and Vasicek (1991) derive the associated credit loss distribution functionwhereas Hartmann-Wendels (2003, pp. 117-120) illustrates the meaning of the IRB risk weights.Gordy (2003) characterizes the necessary and sufficient assumptions that underpin the basic IRBformula. Concerning this interpretation, see also Gordy/Howells (2006, p. 397).

19Cf. Article 272 in connection with Articles 40 and 44, BCBS (2004).


Table 3.6: Constraints by regulation through VaR

This table shows the constraints on the total loan volume in relation to the bank’s initial equityWB if the bank must fulfill a VaR constraint according to (3.3.43). The constraint depends on theprescribed level of confidence p and on the relation between both promised loan redemptions αiLi,i = 1, 2.

Panel A: α1L1 < α2L2 (excluding Case 3)

p feasibility constraint[1− p1 − p2 + q, 1− p2) [p2α2(1− `)− (1− p1)α1`] · L < τWB

[p2α2(1− `)− (1− p1)α1`] · L ≤ τWB

[1− p2, 1− q) and[p1α1`− (1− p2)α2(1− `)] · L < τWB

Panel B: α1L1 = α2L2

p feasibility constraint[1− p1 − p2 + q, 1− p1) (p1 + p2 − 1) · α1L1 < τWB

[1− p1, 1− q) (p1 + p2 − 1) · α1L1 ≤ τWB

Panel C: α1L1 > α2L2 (excluding Case 2)

p feasibility constraint[1− p1 − p2 + q, 1− p1) [p1α1`− (1− p2)α2(1− `)] · L < τWB

[p1α1`− (1− p2)α2(1− `)] · L ≤ τWB

[1− p1, 1− q) and[p2α2(1− `)− (1− p1)α1`] · L < τWB

The level p = 1 − q results in a condition that is always fulfilled if L1, L2,WB ≥ 0

holds with at least one strictly positive value. Hence, condition p = 1− q does not

have an impact on the bank’s behavior.

The bank’s decision problem under regulation by a VaR constraint can be formulated

as follows:20

max`, RD

E[WB(`, RD; j)

]

s.t. ` ∈ [0, 1] (3.3.44)

∃j : (Ds(`, RD; j), `, RD) ∈ Cj

P[L ≤ (p1α1`+ p2α2(1− `)) · (Ds(`, RD; j) +WB) − τ ·WB

]≤ p .

20Buhler/Koziol/Sygusch (2008, p. 149).


We will refer to (`V , RVD; jV ) as the equilibrium, or the bank’s optimal choice

under the VaR approach. Regulation is said to be binding if the maximum of

the unregulated problem, (`∗, R∗D; j∗) does not fulfill the given level of confidence p.

Two remarks concerning the regulatory constraint are in order.

First, strictly speaking, a well-defined solution to Problem (3.3.44) cannot be

guaranteed as the credit-VaR constraint may entail open feasibility sets for some

confidence levels p as indicated by Table 3.6. The openness results from the

cumulative distribution function that is continuous from the right. Thus, for

practical reasons, the maximization problem is altered by adding a slack variable

ε > 0 such that the strict inequalities, as shown in Table 3.6 become weak. Then the

feasibility constraints are compact sets and the existence result in Result 5 carries

over to the case of regulation by a VaR constraint, analogous to the case of regulation

by fixed risk weights, cf. Result 13.

Second, the bank can choose between two constraints to reach a confidence level

of p ∈ [1 − p2, 1 − q), as shown in Panel A of Table 3.6 for the loan allocation

α1L1 < α2L2. Symmetrically, the bank has two such choices if it wishes to allocate

funds such that α1L1 > α2L2 prevails if the confidence level is set to p ∈ [1−p2, 1−q),cf. Panel C. Since the inequalities in Table 3.6 show a tight link between initial equity

and the feasible total loan/deposit volume, binding regulation always reduces the

level of the total loan/deposit volume. As a consequence the bank will choose, if

possible, the constraint for a given level of confidence p that mitigates this volume

reduction most strongly.

Note that for the constraints shown in Table 3.6, Panel A, given p ∈ [1− p2, 1− q),

p2α2(1− `)− (1− p1)α1` > p1α1`− (1− p2)α2(1− `) iff ` <α2

α1 + α2

holds. Likewise, the following ordering holds for constraints given in Table 3.6,

Panel C:

p1α1`− (1− p2)α2(1− `) > p2α2(1− `)− (1− p1)α1` iff ` >α2

α1 + α2

given p ∈ [1 − p1, 1 − q). Because the bank always opts for the weakest constraint

given a compulsory range of feasible loan-allocation rates, the constraints imposed

by a VaR regulation according to (3.3.43) can be summarized by

[v(`, p) + ε(`, p)] · [Ds(`, RD; j) +WB] ≤ τ ·WB (3.3.45)


where v(`, p) is given by

v(`, p) =

p2α2(1− `)− (1− p1)α1`, p ∈ [1− p1 − p2 + q, 1− p2)

p1α1`− (1− p2)α2(1− `), p ∈ [1− p2, 1− q), ` < α2

α1+α2

(p1 + p2 − 1)α1` , p ∈ [1− p1 − p2 + q, 1− q) , ` = α2

α1+α2

p1α1`− (1− p2)α2(1− `), p ∈ [1− p1 − p2 + q, 1− p1)

p2α2(1− `)− (1− p1)α1`, p ∈ [1− p1, 1− q), ` > α2

α1+α2

(3.3.46)

and ε(`, p) defined as

ε(`, p) =

0 if p ∈ [1− p1, 1− q) and ` = α2

α1+α2,

ε > 0 else .

The function ε(`, p) takes a positive, fixed number ε > 0 if the regulatory constraint

would otherwise result in a strict inequality and the feasibility set of the bank’s

optimizing problem would not be compact. This slack variable becomes dispensable

only in one single case, as shown in the formula above.

The function v(`, p) is continuous in ` for all ` ∈ [0, 1] for fixed p. Thus, if it is

binding, the Regulatory Constraint (3.3.45) imposes a continuous bound in addition

to the other ones of Problem (3.3.1) without regulation.

The function v(`, p) is only piecewise differentiable in the loan-allocation rate ` for

fixed p, i.e. it is only differentiable in ` for fixed p within the ranges (0, α1

α1+α2)

and ( α1

α1+α2, 1), respectively. So, differentiability can be violated as one may have

to consider multiple regulatory constraints, which may be the case for two reasons:

first, in Cases 1 and 4, feasible loan-allocation rates range on intervals around α2

α1+α2.

Second, it may be worthwhile for the bank to follow a tighter level of confidence

than it is obliged to do (we provide an example in Chapter 4). Then the feasible

loan-allocation rates may be on either side of α2

α1+α2such that different constraints

must be considered.

If, for given p, the optimal loan-allocation rates never leave the spaces characterized

by Cases 2 or 3, v(`, p) is differentiable in `. Differentiability in ` also holds for

p = 1− p1 if the bank does not wish to comply with a stricter level of confidence.


With these definitions, the Maximization Problem (3.3.44) reads

max`, RD

E[WB(`, RD; j)

]

s.t. ` ∈ [0, 1] (3.3.47)

∃j : (Ds(`, RD; j), `, RD) ∈ Cj

[v(`, p) + ε(`, p)] · [Ds(`, RD; j) +WB] ≤ τ ·WB .

If the bank is regulated by a VaR constraint, the main features of this economy are

summarized by the following result:

Result 23. An equilibrium (`V , RVD; jV ) always exists. If the bank initially has no

equity, WB = 0, if p1 + p2 > 1 holds, and if the level of confidence p is strictly below

1−p1, the bank cannot issue any deposits, Ds(`V , RVD; jV ) = 0. If p ≥ 1−p1, strictly

positive deposit volumes are also feasible under WB = 0.

The feasibility of positive loan and deposit volumes without any initial equity is

surprising at first glance. However, its feasibility can be explained by the Bernoulli

distribution that enables the bank to arbitrarily concentrate credit risk: if the bank

grants the majority of its funds to one of the two firms, the expected repayments from

the whole loan portfolio minus the regulatory fraction of initial equity, E[L]−τ ·WB,

are dominated by the promised loan redemption volume by that very borrower.

Depending on the firm that has been granted the larger loan volume, either a level

of confidence of 1− p1 or 1− p2 is attained. In terms of the loan-allocation rate `,

a level of confidence of 1− p1 is achieved with rather high values of ` which may be

associated with Case 3 in equilibrium. Likewise, for WB = 0, p = 1− p2 implies low

values of ` that may result in Case 2 in equilibrium. Technically, the bank has to

choose (`V , RVD; jV ) such that

λ · [v(`, p) + ε(`, p)] ·Ds(`, RD; j) = 0

holds. Let regulation be binding; that is, that the regulatory constraint holds with

equality. Hence, by Ds(`, RD; j) ≥ 0, Ds(`V , RVD; jV ) = 0 or v(`, p) + ε(`) = 0 must

be fulfilled. As the former leads to E[WB(·)] = 0 because of Ds(`V , RVD; jV ) = 0 and

WB = 0, the bank chooses `V such that v(`V , p)+ ε(`V ) = 0 holds, implying positive

expected wealth, E[WB(·)] > 0.

If WB = 0 and p = 1− p2, the bank allocates its total funds to loans according to

`V =(1− p2)α2 − ε

(1− p2)α2 + p1α1

∈ (0,α2

α1 + α2

) and jV = 2 , (3.3.48)


and, if WB = 0 and p = 1− p1, according to 21

`V =p2α2 + ε

(1− p1)α1 + p2α2

∈ (α2

α1 + α2

, 1) and jV = 3 . (3.3.49)

Given p = 1− p2, `V = 0 implies that regulation is not binding, as v(0, 1− p2) < 0

holds. Symmetrically, `V = 1 results in non-binding regulation at a confidence level

of p = 1 − p1 as v(1, 1 − p1) < 0 holds. In contrast, neither ` = 0 nor ` = 1 can

reach a level of confidence of p = 1− p1 − p2 + q.

If p = 1 − p1 − p2 + q, the regulatory constraint function v(`, p) is always strictly

positive because p1 +p2 > 1 holds due to (3.2.5). Hence, the feasible deposit volume

DV must reduce to zero if WB = 0. If the bank is still meant to comply with the

regulatory constraint for DV = 0, the tightest constraint must be less than τ . For

p = 1 − p1 − p2 + q, the regulatory constraint function v(`, p) attains its minimum

in ` = α2

α1+α2. Thus, the following equality should hold:

(p1 + p2 − 1) · α1α2

α1 + α2

< τ . (3.3.50)

This guarantees that the bank always has the choice of complying with a confidence

level of p = 1− p1− p2 + q. The parameters of the base case, as shown in Table 4.1,

comply with Restriction (3.3.50), while the parameters used in the example shown

in Table 3.7 do not.

3.3.3.2 Characterization of Case 1 if lV = α2

α1+α2Prevails

If p = 1 − p1 − p2 + q is required, the regulatory constraint function attains its

minimum in ` = α2

α1+α2. The bank will allocate its funds to both loans according to

this rate if initial equity is sufficiently scarce. As a consequence, it will favor the

less risky loan. In this paragraph, we take a closer look at Case 1.

Result 24. Let p ∈ [1 − p1 − p2 + q, 1 − p1). Suppose that regulation is binding,

that the household is not constrained by its initial wealth, and that Case 1 with

21The optimal interest rates RVD can be represented analytically, too, but their expressions arevery lengthy and signs of sensitivities and order of magnitudes remain opaque. This is even still aproblem if the first-order conditions are solved for RD for fixed ` as ∂Ds(·)

∂` and ∂Ds(·)∂RD

· ∂RVD

∂` will

typically exhibit opposite signs making thus the effect of ∂`V

∂θ unclear.


`V = α2

α1+α2prevails. Then the deposit interest rate is

RVD(1) =

q[ε+(p1+p2−1)

α1α2α1+α2

]+q(p1+p2−2q)

[2τ−ε−(p1+p2−1)

α1α2α1+α2

]α1α2α1+α2

γWB −√QV1 (`V ,p)

2q(1−q)[τ−ε−(p1+p2−1)

α1α2α1+α2

]γWB

(3.3.51)

where

QV1 (`V , p) = q2

[ε+ (p1 + p2 − 1) α1α2

α1+α2

]2

+

+ 2q[ε+ (p1 + p2 − 1) α1α2

α1+α2

]·

·q(p1+p2−2q)

[ε+

(p1+p2−1)α1α2α1+α2

]α1α2α1+α2

−2[(1−q)Rf−

(p1+p2−2q)α1α2α1+α2

][τ−ε− (p1+p2−1)α1α2

α1+α2

]γWB

+(p1 + p2 − 2q)

(p1+p2−2q)q[ε+

(p1+p2−1)α1α2α1+α2

]2−4(1−p1−p2+q)τ

[τ−ε− (p1+p2−1)α1α2

α1+α2

] α21α

22γ

2W 2B

(α1+α2)2.

The associated deposit volume amounts to

DV =(α1 + α2)(τ − ε)− (p1 + p2 − 1)α1α2

(α1 + α2)ε+ (p1 + p2 − 1)α1α2

·WB . (3.3.52)

The signs for the sensitivities of the deposit interest rate RVD are given as follows,

∂RVD

∂WB

> 0,∂RV

D

∂γ> 0,

∂RVD

∂Rf

> 0,∂RV

D

∂τ> 0 . (3.3.53)

We note that the equilibrium examined above is not the only possible

characterization of an equilibrium under Case 1 if VaR regulation at this confidence

level is binding. We provide examples in Chapter 4: Figure 4.4 shows that `V

deviates from α2

α1+α2to catch up with `∗. As a result, the total loan volume exhibits

a kink and its pro-cyclical behavior slackens (Fig. 4.2).

Beyond Case 1, other Cases may prevail under this level of confidence, also if WB > 0

holds. So, the numerical studies in Chapter Chapter 4 provide examples for Case 2

to occur. Figure 4.22 illustrates that `V may drift to zero, which causes in turn a

migration from Case 1 to Case 2 and jumps in total lending, as Figure 4.20 shows.

The reason for RVD increasing in WB is as follows: because the loan-allocation rate

`V and the deposit-supply function itself are independent of WB, the interest rate

RVD is the only variable the bank can use to steer the deposit volume with respect

to changes in WB. Furthermore the overall deposit volume DV strictly increases in

WB, as does the deposit-supply function in RVD according to Result 5. A similar

mechanism is at work in the equilibria presented by Result 14 and 21. In particular,

this argument applies to those equilibria under VaR regulation where both firms’

projects are equal and where `V = 12

(cf. Results 25 and 28).


The derivations of the formulæ are presented in Appendix A.2.3.2.

According to (3.3.52) the total loan volume increases linearly with a slope exceeding

one,∂LV

∂WB

=(α1 + α2) · τ

(α1 + α2)ε+ (p1 + p2 − 1)α1α2

> 1 .

Since the loan-allocation rate does not depend on the bank’s initial equity, the

relation between the total loan/deposit volume and WB simplifies to a simple linear

relation as in the case under regulation with fixed risk weights and equality in the

firms’ projects, see Result 15. But it remains unclear if shocks in WB affect the total

loan/deposit volume in a pro-cyclical manner as the monotonicity behavior of the

loan/deposit volume in the laissez-faire equilibrium remains opaque under equally

general conditions.

Concerning shocks in the success probabilities pi and the gross interest rates on

loans, αi, there is a potential for counter-cyclical effects as the total loan volume

under regulation reacts upon those shocks by

∂LV

∂pi= − α1α2(α1 + α2) · τ ·WB

[(α1 + α2)ε+ (p1 + p2 − 1)α1α2]2< 0 , i = 1, 2 ,

∂LV

∂αi= − (p1 + p2 − 1) · α2

3−i · τ[(α1 + α2)ε+ (p1 + p2 − 1)α1α2]2

< 0 , i = 1, 2 .

The same signs prevail if common shocks to pi and αi, respectively, are considered,

∂LV

∂mp

= (p1 + p2) · ∂LV

∂pi, and

∂LV

∂mα

= (α1 + α2) · αiα3−i

· ∂LV

∂αi, i = 1, 2 ,

where mp is the common multiplier for pi and mα the common multiplier concerning

αi, i ∈ 1, 2, respectively.

That higher success probabilities and higher gross returns on the firms’ projects

(i.e. higher loan interest rates) lessen the optimal total loan volume seems surprising

at first glance but is actually tautological: the higher pi or the higher αi become,

the tighter the regulatory constraint becomes, implying in turn this property. These

effects can thus be explained via the sensitivities of

v(α2

α1 + α2

, 1− p1 − p2 + q) = (p1 + p2 − 1)α1α2

α1 + α2

.

A rise in pi or in αi always translates into a rise of v(·), thus making the regulatory

constraint stricter and making the maximum feasible loan/deposit volume smaller.

This simple relation is particularly due to the simple formula for the optimal loan-


allocation rate that strictly increases in each αi and does not depend on any other

parameter.

Furthermore, note that we have implicitly assumed for the above argument that the

level of confidence p is kept structurally fixed for changes in pi, in the sense that p

still complies with a confidence level from [1− p1 − p2 + q, 1− p1) after changes in

the probabilities pi. That is, movements in pi are assumed to be tight enough such

that the level of confidence p does not leave its initial range.

The total loan volume, LV , increases in τ as an increasing τ means a relaxing

regulatory constraint: the higher is this multiplier, the higher unexpected losses can

become that are tolerated at a fixed level of confidence p according to (3.3.45), i.e.

∂LV

∂τ=

(α1 + α2) ·WB

[(α1 + α2)ε+ (p1 + p2 − 1)α1α2]2> 0 .

By the binding regulation and by the fact that loans are granted such that their

promised redemptions are equal to each other, the household’s trade-off between risk

and return loses its importance. Thus, the total loan/deposit volume does neither

depend on its absolute risk aversion γ nor on the gross return on the risk-free asset

Rf . We note that this argument does not hold in general: especially if the bank’s

initial equity exceeds a given threshold under Case 1, the equilibrium loan-allocation

rate may depend on other parameters than the productivity parameters as outlined

in Result 24. In this respect we refer to the examples, as shown by Figures 4.4 and

4.22.

As shown by Result 24, there is a complementary effect between the risky deposits

and the risk-free asset which is reflected by

∂RVD(1)

∂Rf

> 0 . (3.3.54)

The interest rate offered by the bank strictly increases in the rate on the risk-free

asset since for the respective sensitivity of the total deposit volume

∂DV

∂Rf︸︷︷︸=0

≡ ∂Du(`V , RVD(j); j)

∂Rf︸︷︷︸<0

+∂Du(`V , RV

D(j); j)

∂RD︸︷︷︸>0

·∂RVD(j)

∂Rf

= 0 , j ∈ 1, 4 ,

holds true. The positive sign of the partial derivative of Du with respect to RD

is due to the assumed optimality of the choice, as shown by Result 24, cf. in this

respect Result 5.


To gain further insights into the sensitivities of total loan volumes under the VaR

regulation, we consider the case of identically distributed project returns. This

approach allows us to draw comparisons with the sensitivities of the respective

unregulated total loan volumes and we can judge whether regulation affects lending

in a pro-cyclical way or not. Result 24 becomes:

Result 25. Let p1 = p2 = p and α1 = α2 = α. Assume that 2(τ + ε) > (2p − 1)α

holds. Suppose Case 1 with `V = 12

is optimal, that regulation is binding, and

that the household is not constrained by its initial wealth. Furthermore, let p ∈[1− 2p+ q, 1− p). Then the bank promises

RVD =

q·[(2p−1)α+2ε]·[1−(p−q)αγWB ] + 4q(p−q)αγ·τWB −√QV1 (p,q,α)

2q(1−q)[2(τ−ε)−(2p−1)α]γWB


QV1 (p,q,α) = q2·(2p−1)α+2ε+(p−q)α[2(2τ−ε)−(2p−1)α]γWB2

− 4q(1−q)γWB [2(τ−ε)−(2p−1)α]·[(2p−1)α+2ε][Rf−(p−q)α]+(p−q)(1−2p+2q)α2γτWB .

Furthermore, if

τ ≥[(2p− 1)α + 2ε]

[(p− q)2α2 − (p− q)(2− q)αRf + (1− q)R2

f

]

(p− q)(2p− q)α2 − 4(p− q)αRf + 2(1− q)R2f

holds, the interest rate on deposits strictly increases in the bank’s initial equity,

∂RVD

∂WB

> 0 .

The optimal deposit volume is given by

DV =[2(τ − ε)− (2p− 1)α]WB

(2p− 1)α + 2ε.

If both firms are equal, some basic comparisons can be drawn between the outcome

when the bank is unregulated and when it is regulated by a VaR approach:

Result 26. Suppose that the assumptions of Result 6 and 25 hold. Then regulation

makes the total loan volume, L, pro-cyclical concerning changes in τ , i.e. the

regulated total loan volume, LV , strictly increases. But the regulated total loan

volume is neither pro-cyclical concerning shocks in γ nor shocks in Rf . Furthermore,

regulation results in a lower interest rate on deposits, RVD < R∗D. Assume

furthermore α ≤ 43. Then regulation makes the total loan volume, L, on average


pro-cyclical concerning shocks in WB.

The proof to this result is analogous to the proof to Result 15. The effects of

regulation on the total loan/deposit-volume are straightforward since the respective

sensitivities under regulation are given by

∂LV

∂τ=

2

[(2p− 1)α + τ ]> 0

∂LV

∂γ= 0 =

∂LV

∂Rf

.

Except for these parameters, no further conclusions on pro- or counter-cyclical effects

by credit-VaR can be drawn as the sensitivities of the total loan volume to any other

shocks are unknown in the case of the unregulated bank. Given the assumptions

of Result 25, the remaining sensitivities of the regulated total loan volume are as

follows:

∂LV

∂p= − 4ατ

[(2p− 1)α + τ ]2< 0 ,

∂LV

∂α= − 2(2p− 1)τ

[(2p− 1)α + τ ]2< 0 .

The reasons for these signs are exactly the same reasons that have been put forward

in connection with Result 24 as Result 25 represents just a special case of the former.

Again, Case 1 does not need to be the optimal Case under binding VaR regulation

when both firms are equal as the following example illustrates:


As under regulation with fixed risk weights, Case 4 allows for equilibria with an

asymmetric allocation of promised loan redemptions. These solutions are supported

by appropriate levels of confidence:

Result 27. Suppose that regulation is binding,that the household is not constrained

by its initial wealth, that WB > 0, and that Case 4 is optimal. Let p ∈ [1−p1, 1−p2).

Then

α1LV1 > DVRV

D , α2LV2 = DVRV

D ,


`V =(α2 −RV

D)(q4RVD −Rf ) + q4(1− q4)γWBα2

(RVD

)2

[q4RV

D −Rf + q4(1− q4) (RVD)

2γWB

]α2

,

(3.3.55)

RVD =

[(1− p1)α1 + p2α2]Rf + q4 [(1− p1)α1 − ε]α2 +√QV

4 (1− p1)

2q4 (1− p1)α1 + p2α2 − (1− q4)α2 [(1− p1)α1 + τ − ε] γWB,

where

QV4 (1− p1) = [(1− p1)α1 + p2α2]Rf − q4α2 [(1− p1)α1 − ε]2

+ 4q4(1− q4)α22 · [(1− p1)α1 − ε] · [(1− p1)α1 + τ − ε] γWBRf

is feasible in equilibrium.

If p ∈ [1− p2, 1− q), then

α1LV1 = DVRV

D , α2LV2 > DVRV

D ,

`V =(q4R

VD −Rf )R

VD[

q4RVD −Rf + q4(1− q4) (RV

D)2γWB

]α1

,

(3.3.56)

RVD =

[p1α1 + (1− p2)α2]Rf + q4 [(1− p2)α2 − ε]α1 +√QV

4 (1− p2)

2q4 p1α1 + (1− p2)α2 − (1− q4)α1 [(1− p2)α2 + τ − ε] γWB,

where

QV4 (1− p2) = [p1α1 + (1− p2)α2]Rf − q4α1 [(1− p2)α2 − ε]2

+ 4q4(1− q4)α21 · [(1− p2)α2 − ε] · [(1− p2)α2 + τ − ε] γWBRf

is feasible in equilibrium. If WB = 0 neither (3.3.55) nor (3.3.56) can occur as

equilibrium.

Technically, those formulæ can be obtained by first solving the Case constraint

for the loan-allocation rate that is assumed to bind. Second, the VaR constraint

is solved for RD. Here, two solutions arise whereas one cannot be optimal as it

exceeds the other by far, cf. Result 5. If WB = 0 holds, the optimal deposit interest

rates presented in Result 27 imply that the bank can no longer accept deposits: the

deposit volume goes to zero as RVD = 1

q4Rf holds for WB = 0 and the loan-allocation

rate is indeterminate. Then potential equilibria may be characterized by (3.3.48) or

(3.3.49), respectively.

If both equilibria are compared with the laissez-faire Equilibrium (3.3.11), regulation


affects the total loan/deposit volume and the deposit interest rate in a pro-cyclical

way concerning shocks in WB. The reason is that the respective unregulated

outcomes do not depend on WB, while the regulated outcomes do, in particular

at both levels of confidence.

Both presented potential equilibria indicate (such as the extreme loan-allocation

rates given by (3.3.48) and (3.3.49) if WB = 0) that intermediate levels of confidence,

i.e. p ∈ [1− p1, 1− q), support loan allocations that tend to one single loan. This

property arises since a given level of confidence is strongly related to the failure

probability of a given loan due to the Bernoulli distribution.

Let us provide some deeper intuition as to why such loan-allocation rates may be

optimal for the bank owners when regulation is binding. Consider first a level

of confidence of p = 1 − p2. If the bank owners allocate their funds such that

α2L2 < α1L1 holds, the VaR Constraint (3.3.45) requires the bank to comply with

E(L)− τ ·WB

!< α1L1 .

But by turning the loan allocation once around, i.e. choosing α1L1 < α2L2, the

bank still complies with the (1− p2)-level as it fulfills now

E(L)− τ ·WB

!< α2L2 .

By doing so, the bank owners can raise their expected return on their loan portfolio,

thus raising their overall expected wealth if the total promised deposit redemption

DV ·RVD does not increase too much.

Consider now a level of confidence of p = 1 − p1. If the bank owners allocate their

funds such that α1L1 < α2L2 holds, the VaR Constraint (3.3.45) requires that the

bank complies with

E(L)− τ ·WB

!< α1L1 ,

i.e. to the (1− p1− p2 + q)-level because of 1− p1− p2 + q < 1− p1 < 1− p2. Again,

by reversing the loan allocation, i.e. choosing α2L2 < α1L1, the bank still complies

with the (1− p1)-level while lifting up the constraint to the higher loan redemption

volume; that is, say to

E(L)− τ ·WB

!< α1L1 .

Thus, by doing so, the bank owners can raise their business volume and thus mitigate

the negative size effect on their business induced by regulation. Furthermore, deposit

interest rates will not rise, as long as the loan portfolio’s risk and the risk of deposits


has declined.

Both arguments are also reflected by the regulatory function v(`, p) given by (3.3.46):

v(`, 1− p2) has its unique maximum in ` = α2

α1+α2suggesting that the bank chooses

`V < α2

α1+α2to mitigate the reduction in the loan/deposit volume while not forgoing

profits as it would be the case if it chose `V > α2

α1+α2. This notion is illustrated by

the example given in Table 3.7.

This example illustrates that the regulatory constraint affects the loan-allocation

rate depending on the required level of confidence. If p ∈ [1− p1, 1− p2), the bank

chooses loan-allocation rates above α2

α1+α2. At this level of confidence, Case 2 is not

feasible under this parametrization: Case 2 requires the bank to comply with

[p2α2(1− `)− (1− p1)α1`+ ε] · [Ds(`, RD; 2) +WB] ≤ τWB .

As v(`, 1 − p2) reaches its minimum in ` = α2

α1+α2, the bank can minimize the

regulatory constraint just at this point given fixed values of the deposit interest

rate. But the parametrization, as shown in Table 3.7 results in

v(α2

α1 + α2

, 1− p2) = v(0.50099, 0.2) = 0.380111 > 0.1 = τ .

Hence, there is no feasible solution under Case 2 unless regulation is dispensed with

for DV = 0 under Case 2. Yet, there is always a non-trivial solution under Case 3

for p = 1 − p1 because v(1, 1 − p1) = −(1 − p1)α1 < 0 holds. Then the bank is in

fact not restricted in its choice under regulation given Case 3, as shown in Table 3.7,

Panels B and D. But in fact the choice compatible with Case 4 is the equilibrium.

It is characterized by (3.3.55). In the examples discussed in the following chapter,

Chapter 4, Case 4 does not emerge. Without regulation, the equilibrium is given by

the bank’s optimal choice given Case 2. Above all, both the bank and the household

are in fact better off without regulation.

Likewise, the bank chooses loan-allocation rates below α2

α1+α2if a confidence level of

p ∈ [1− p2, 1− q) must be met. Here, the solution given Case 2 is the equilibrium

under regulation. By v(0, 1− p2) = −(1− p2)α2 < 0 the bank can freely choose its

deposit interest rate and deposit volume. By incidence, the regulated equilibrium

coincides with the laissez-faire equilibrium.

It is striking that the deposit interest rates under Case 1 both with and without

regulation exceed the respective average gross return on loans, α1`+α2(1− `). This

is the case as any other solutions with RD(1) below α1`+ α2(1− `) are not feasible

under Case 1: the deposits volumes become too low so that the Case constraints are


Table 3.7: Example illustrating Result 27

The upper panel reports parameter values used in this example. Panel B reports the bank’s optimalchoices under regulation with VaR at a confidence level of p ∈ [1− p1, 1− p2), Panel C the bank’soptimal choices under VaR at a confidence level of p ∈ [1− p2, 1− q). The lower panel shows thebank’s optimal choices without regulation.


p1 p2 q α1 α2 WB γ WH Rf τ ε

0.802 0.8 0.79 1.25 1.255 50 0.001 3000 1 0.1 0.0001

Panel B: Optimal solutions under regulation with VaR at a confidence level ofp ∈ [1− p1, 1− p2)

`V (j) RVD(j) j DV (·, ·; j) E[WVB (·, ·; j)] U(WH) Eq./Res.

0.783236 1.29961 1 157.782 45.0104 3003.98 n/an/a n/a 2 n/a n/a n/a n/a1 1.25341 3 20.9745 50.6369 3000.05 n/a

0.743814 1.23663 4 17.7565 50.6649 3000.04 (3.3.55)

Panel C: Optimal solutions under regulation with VaR at a confidence level ofp ∈ [1− p2, 1− q)

`V (j) RVD(j) j DV (·, ·; j) E[WVB (·, ·; j)] U(WH) Eq./Res.

0.219244 1.30158 1 157.307 45.2417 3003.92 n/a0 1.25746 2 23.5749 50.7423 3000.07 Res. 81 1.25341 3 20.9745 50.6369 3000.05 n/a

0.258458 1.23667 4 17.8734 50.7139 3000.04 (3.3.56)

Panel D: Optimal solutions without regulation

`∗(j) R∗D(j) j D∗(·, ·; j) E[W ∗

B ] U(W ∗H) Eq./Res.

0.50099 1.26343 1 49.9257 49.8315 3000.99 n/a0 1.25746 2 23.5749 50.7423 3000.07 Res. 81 1.25341 3 20.9741 50.6369 3000.05 n/a

0.304016 1.23798 4 22.4032 50.7188 3000.06 (3.3.11)


violated. Irrespective of a specific parametrization chosen and the bank’s behavior

under the other Cases, this choice can never be an equilibrium as it will be always

dominated by the choice ` = 0 under autarky, resulting in p2α2WB, here in numbers

50.6 (` = 0 is due to (3.2.5)). This choice is in turn always dominated by the choice

(`(j), RD(j); j) = (0, RD(0; 2); 2), here resulting in 50.6369. Note that without

regulation, (0, RD(0; 2); 2) is always feasible.

To sum up, Case 3 may rather arise in equilibrium if p ∈ [1− p1, 1− p2) is required,

while Case 2 may rather occur if p ∈ [1 − p2, 1 − q) is set. So, in equilibrium,

changes in the Case are very likely to occur for intermediate levels of confidence.

Specifically, Case 2 will rather dominate Cases 1 and 4 for p = [1− p2, 1− q) as the

higher expected return on the second loan is an incentive to the risk-neutral bank

owners to shift the loan-allocation rate far enough to reach Case 2. By the lower

expected return on the first loan, the incentive to deviate from either Case 1 or 4

in favor of Case 3 is not that strong for p = [1 − p1, 1 − p2). Figure 4.3 further

illustrates these incentives induced by the different levels of confidence.

If both firms’s project returns and risks are equal, Case 4 may prevail as equilibrium

if a confidence level of p = 1− 2p+ q is set. It can be characterized as follows:

Result 28. Let p1 = p2 = p and α1 = α2 = α. Suppose Case 4 is optimal, that

regulation is binding, and that the household is not constrained by its initial wealth.

Furthermore, let p ∈ [1 − 2p + q, 1 − p). Then the bank chooses `V = 12

as loan-

allocation rate and promises

RVD =

(2p− q) [(2p− 1)α + 2ε] −√QV

4 (p, q, α)

2(2p− q)(1− 2p+ q) [2(τ − ε)− (2p− 1)α] γWB


QV4 (p, q, α) = (2p− q)2 [(2p− 1)α + 2ε]2

− 4(2p− q)(1− 2p+ q) [2(τ − ε)− (2p− 1)α] · [(2p− 1)α + 2ε] γWBRf .

The optimal deposit volume is given by

DV =[2(τ − ε)− (2p− 1)α]WB

(2p− 1)α + 2ε, (3.3.57)

resulting in the following signs for the sensitivities

∂DV

∂WB

> 0,∂DV

∂γ= 0,

∂DV

∂Rf

= 0,∂DV

∂p< 0,

∂DV

∂q= 0,

∂DV

∂α< 0,

∂DV

∂τ> 0 ,

(3.3.58)


in turn implying the signs for the sensitivities of the deposit interest rate RVD,

∂RVD

∂WB

> 0,∂RV

D

∂γ> 0,

∂RVD

∂Rf

> 0,∂RV

D

∂p< 0,

∂RVD

∂q> 0,

∂RVD

∂α< 0,

∂RVD

∂τ> 0 .

(3.3.59)

By p1 = p2 and α1 = α2, the bank’s objective function, E[WB(`, RD; 4)

], is

independent of the loan-allocation rate `. Assume furthermore, that the Case

constraints are not violated, i.e. that the bank will choose (`V (4), RVD(4)) from the

interior. Then the Lagrangian depends on ` only via the regulatory constraint

function v(`, 1− 2p+ q), which is

v(`, 1− 2p+ q) =

(p− `)α if ` ≤ 1

2,

pα− α(1− `) if ` > 12.

Consequently, the bank first chooses ` such that the regulatory burden is minimized,

resulting in `V = 12. Obviously, the choice is compatible with Case 4 for any DV ≥ 0.

RVD results from the regulatory constraint if it holds with equality, as assumed.

Concerning the principle of the derivation of the interest rate’s sensitivities, we refer

to Result 21. The deposit interest rate strictly decreases in α as with `V = 12, RD is

the only variable left to the bank to decrease the total loan/deposit volume while the

regulatory function v(12, p) strictly increases. The deposit-supply function Ds(RD; 4)

does not depend on α in Case 4. The remaining sensitivities can be explained along

the lines of the discussion on p. 77 following Result 11.

Comparing the equilibrium given by Result 28 with the laissez-faire equilibrium

characterized by Result 11 allows for the following conclusions concerning cyclical

impacts:

Result 29. Suppose that the assumptions of Results 11, and 28 hold. Then

regulation makes the total loan volume, LV , and the deposit interest rate RVD pro-

cyclical concerning changes in τ and regulation results in a smaller interest rate on

deposits, RVD < R∗D. But LV is not pro-cyclical concerning shocks in γ, Rf , and q.

Let (3.3.13) be the reference equilibrium without regulation. Then the regulated total

loan volume is pro-cyclical on average concerning shocks in WB.

If (3.3.15) is the reference equilibrium without regulation, the regulated total loan

volume is pro-cyclical concerning shocks in WB, but is counter-cyclical concerning

shocks in p and α. The regulated deposit interest rate RVD is pro-cyclical concerning

changes in WB and γ, but is counter-cyclical concerning changes in α.


Table 3.8: Example of equal projects under VaR-based regulation,Bernoulli modelThe upper panel shows an example set of parameter values with equal success probabilities and equalproductivity parameters for both projects. The lower panel reports the bank’s optimal choices in eachCase j = 1, . . . , 4. The equilibrium under VaR regulation at a confidence level of p = 1−p1−p2 +q

is characterized by Case 4. Without regulation, the equilibrium is given under Cases 2 and 3,respectively, cf. Table 3.3.


p q α WB γ WH Rf τ ε

0.92 0.9 1.15 10 0.001 3000 1.05 0.99 0.0001


`V (j) RVD(j) j DV (·, ·; j) E[WB(·, ·; j)] U(WH) Eq./Res.12 1.14195 1 10.4927 10.426 3150.23 Res. 25

0.498618 1.12965 2 10.4255 10.541 3150.12 n/a0.501382 1.12965 3 10.4255 10.541 3150.12 n/a

12 1.11781 4 10.4927 10.6562 3150 Res. 28

The cyclical impacts can be read-off by the signs of the sensitivities stated in Results

11, and 28.

Let us revisit the example shown in Table 3.3. Table 3.8, Panel B, reports the bank’s

optimal choices if it must comply with a confidence level of p = 1 − p1 − p2 + q.

The regulatory parameter τ is set equal to 0.99 such that regulation just becomes

binding under Case 4. The equilibrium under regulation is characterized by Case 4.

Both, under Case 1 and 4, the regulated bank can issue the same volume of deposits.

But the risks the household must bear under these two Cases are different in the

sense that the probability of full deposit redemption differs. As a consequence, the

deposit interest rate is lower under Case 4 than under Case 1, resulting in total in

a higher expected final wealth for the bank under Case 4.

If τ = 1, regulation is effectively binding in Cases 1 to 3, too, while in Case

4, regulation just becomes binding such that the bank’s choice is identical to its

choice without regulation, cf. Table 3.3. Then the equilibrium under regulation is

characterized by the bank’s choice according to Case 4 as well.


3.4 Summary

In this chapter, a three-sector model of financial intermediation was introduced and

analyzed. Some specific equilibria, distinguished by Case and by whether both

firms’ projects are different in return and risk or not, have been characterized.

Explicit expressions for the loan-allocation rate, the deposit interest rate, and the

total loan/deposit volume have been obtained. These characterizations led to first

insights concerning the issue of whether risk-sensitive capital adequacy rules may

enhance or even create cyclical patterns in lending. Results 15, 17, 20, 22, 26, and 29

give first answers. They have been summarized in Table 3.9 concerning an approach

of fixed risk weights and in Table 3.10 concerning a VaR approach.

Shocks in the bank’s initial capital, reflecting past gains or losses, seem to affect

lending in a pro-cyclical manner. This observation also applies to the deposit interest

rate. Shocks in the depositor’s risk-aversion, however, have little or no impact on

regulated outcomes. Changes in the success probabilities pi and q or marginal gross

returns on loans, α, have mostly no pro-cyclical effect. In contrast, we observe

counter-cyclical effects concerning shocks in pi and αi under the VaR regulation.

The specific instance is characterized by Case 4 and by projects that have the same

success probability and gross return. Clearly, changes in the regulatory parameters

affect the optimal loan volumes and deposit interest rates under a binding regulation

so that there are pro-cyclical effects in this respect by definition.

The results so far address few specific equilibria only so that there are many open

questions left. In particular, in most equilibria considered and compared, the loan-

allocation rate remains the same, no matter if it is the laissez-faire or the regulated

equilibrium. Hence, we have not considered any effects concerning the allocation of

risks. These effects, in turn, could have strong impacts on the total loan/deposit

volume and on deposit interest rates and thus alter their cyclical behavior.

Furthermore, the equilibria considered have not been fully characterized concerning

all potential pro-cyclical effects. For many shocks the issue of pro-cyclicality remains

unsolved, even if the signs of the respective sensitivities with and without regulation

are known. But then, their absolute magnitudes remain opaque or some further

strong and specific assumptions are required.

3.4. SUMMARY 115

Tab

le3.

9:Sum

mary

of

cycl

ical

eff

ect

sth

rough

regula

tion

by

fixed

risk

weig

hts

Thi

sta

ble

sum

mar

izes

the

cycl

ical

effec

tsth

atha

vebe

enth

eore

tica

llych

arac

teri

zed

inSe

ctio

n3.

3.2.

The

diff

eren

tno

tion

sof

cycl

ical

ity

are

base

dup

onth

ede

finit

ions

(2.2

.1)

to(2

.2.4

).T

hela

stro

wad

dres

ses

som

esp

ecifi

cvo

lum

eeff

ects

.

Cas

e1

Cas

e2

Cas

e4

equi

libri

ump

1=p

2,α

1=α

2,

c 1<c 2

p16=p

2,α

16=α

2p

1=p

2,α

1=α

2

unde

rc 1

=c 2

`S=

0(3

.3.3

4),

(3.3

.36)

(3.3

.39)

c 1=c 2

regu

lati

onR

esul

t14

Res

ult

16R

esul

t19

Res

ult

19R

esul

t21

refe

renc

ep

1=p

2p

16=p

2,α

16=α

2p

1=p

2,α

1=α

2eq

uilib

rium

α1

=α

2`∗

=0

(3.3

.10)

,(3

.3.1

1)(3

.3.1

3)(3

.3.1

5)w

/ore

gula

tion

Res

ult

6R

esul

t8

Res

ult

9R

esul

t11

Res

ult

11cy

clic

aleff

ects

stat

edin

Res

ult

15R

esul

t17

Res

ult

20R

esul

t22

sens

itiv

itie

sLS

RS D

LS

RS D

LS

RS D

LS

RS D

LS

RS D

LS

RS D

WB

pro-

cycl

ical

onav

erag

eifα≤

4 3

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

onav

-er

age

pro-

cycl

ical

onav

-er

age

pro-

cycl

ical

γ Rf

n/a

n/a

n/a

n/a

n/a

p1

not

pro-

cycl

ical

n/a

not

pro-

cycl

ical

not

pro-

cycl

ical

not

pro-

cycl

ical

not

pro-

cycl

ical

n/a

p2

n/a

q α1

not

pro-

cycl

ical

not

pro-

cycl

ical

not

pro-

cycl

ical

not

pro-

cycl

ical

α2

not

pro-

cycl

ical

n/a

cpr

o-cy

clic

alpr

o-cy

clic

alc 1

pro-

cycl

ical

pro-

cycl

ical

not

pro-

cycl

ical

not

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

pro-

cycl

ical

c 2pr

o-cy

clic

alpr

o-cy

clic

alab

solu

teLS

=1cc1·W

B,

LS

=1cc2·W

B,

valu

esRS D<R∗ D

RS D<R∗ D

RS D<R∗ D

LS

=1cc1·W

B,RS D<R∗ D


Tab

le3.10:

Sum

mary

of

cyclica

leff

ects

by

VaR

-base

dre

gula

tion

This

tablesum

marizes

thecyclicaleff

ectsthat

havebeen

theoreticallycharacterized

inSection

3.3.3.T

hediff

erentnotions

ofcyclicality

arebased

uponthe

Definitions

(2.2.1)to

(2.2.4).T

helast

rowaddresses

some

specificvolum

eeff

ects.

Case

1C

ase4

equilibriump

1 6=p

2 ,α

1 6=α

2p

1=p

2 ,α

1=α

2p

1 6=p

2 ,α

1 6=α

2p

1=p

2 ,α

1=α

2under

p=

1−p

1 −p

2+q

p=

1−2p

+q

p=

1−p

1p

=1−

p2

p=

1−2p

+q

regulationR

esult24

Result

25R

es.27:

(3.3.55),(3.3.56)

Result

28reference

p1 6=

p2

p1

=p

2p

1 6=p

2 ,α

1 6=α

2p

1=p

2 ,α

1=α

2equilibrium

α1 6=

α2

α1

=α

2(3.3.10)

(3.3.11)(3.3.13)

(3.3.15)w

/oregulation

n/aR

esult6

Result

9R

esult9

Result

11R

esult11

cyclicaleffects

p.107

follow-

discussedby/on

pp.102

-104

Result

26n/a

ingR

es.27

Result

29

sensitivitiesLV

RVD

`V

LV

RVD

LV

RVD

LV

RVD

LV

RVD

LV

RVD

WB

pro-cyclicalonaverage

pro-cyclicalonaverage

pro-cyclicalpro-cyclicalonaverage

pro-cyclical

pro-cyclical

γnotpro-

n/anotpro-

notpro-

cyclicaln/a

notpro-

notpro-

Rf

cyclicalcyclical

n/an/a

cyclicaln/a

cyclicalpi

n/an/a

n/an/a

n/acounter-cyclical

n/a

qnotpro-cyclical

notpro-

cyclicalnotpro-cyclical

notpro-cyclical

αi

n/an/a

n/acounter-cyclical

counter-cyclical

τpro-cyclical

pro-cyclical

pro-cyclical

pro-cyclical

pro-cyclical

pro-cyclicalpro-cyclical

pro-cyclical

pro-cyclical

pro-cyclical

pro-cyclical

absoluteLV

=τ

(p1+p2 −

1)α

1`

V+ε ·W

B,

LV

=2τ

(2p−

1)α

+2ε ·W

B,

LV

=2τ

(2p−

1)α

+2ε ·W

B,

values`V

=α

2α

1+α

2`V

=` ∗

=12 ,RVD<R∗D

`V

=` ∗

=12 ,RSD<R∗D

Chapter 4

Numerical Analysis of Regulatory

Impacts

4.1 Introduction

So far general results concerning the existence of an equilibrium allocation and

some results concerning some specific equilibrium outcomes were obtained. There

was only partial evidence of pro-cyclicality and other potential effects induced by

regulation. Results 15, 17, 20, 22, 26, and 29 provide first notions on how regulation

can affect loan and deposit volumes depending on the state of the cycle. Particularly,

changes in the bank’s initial equity to which loan volumes are ultimately pegged

seem to affect total loan volumes in a qualitatively different manner than changes in

expected prospects. However, as Result 17 is characterized by a single loan granted

in equilibrium and as Results 15, 22, 26, and 29 are based on the assumption of

both firms undertaking equal projects, the typical properties of risk-based capital

regulation cannot come into effect. For this reason, we will now study the model

numerically in order to account for differences in the expected returns and in the

return volatilities of the firms’ projects. Thus there is a serious role for risk-sensitive

capital regulation and equilibrium outcomes will depend to a greater extend on the

differing prospects of the firms’ projects, on the differentiated treatment of these

projects by regulation, and on the household’s risk attitude who is the ultimate

investor in this model.

Furthermore, the numerical analysis may provide deeper insights into regulatory

effects as it can explicitly account for migrations from one Case to another due

to regulation. The results shown so far have always compared an outcome under

regulation to a given laissez-faire outcome without questioning the relevance of the

117

118 CHAPTER 4. NUMERICAL ANALYSIS OF REGULATORY IMPACTS

Table 4.1: Parameter values of the base case, Bernoulli model

The upper panel reports the parameter values used for this numerical analysis of the model setforth in Chapter 3. The lower panel reports the thus implied values of the expected gross returnson single loan redemptions and the variances of these returns. Furthermore, the correlation of bothloans’ returns are shown.

Panel A:

p1 p2 q α1 α2 WB γ WH Rf τ ε

0.995 0.99 0.987 1.15 1.2 100 0.008 3000 1.05 4 0.000001

Panel B:

E[L1/L1] E[L2/L2] V[L1/L1] V[L2/L2] Corr[L1/L1, L2/L2]= 1.14425 = 1.188 ≈ 0.0811137 ≈ 0.119398 ≈ 0.277856

Cases thus considered. As a by-product, it becomes more meaningful in this vein

to point out other regulatory effects, notably those on the loan allocation. This

may complement the discussions initiated by Results 18 and 20, or by the equilibria

under the VaR approach characterized by (3.3.48) and (3.3.49), respectively.

The base case scenario is given by the parameter values outlined in Table 4.1.

Parameter values are chosen such that Conditions (3.2.5), (3.2.6), and (3.2.7) are

fulfilled. The expected return, variance of returns, and the correlation of the loans

granted to both firms are identical to those of the corresponding projects as implied

by the clearing of the loan market (3.2.4).1

As the default probabilities of the loans amount to 0.5% and 1.0%, respectively,

credit grades of BBB and BB+ apply if these probabilities are considered as

historical one-year default probabilities.2 Such credit grades translate into a

1Buhler/Koziol/Sygusch (2008) study the model outlined in Chapter 3 by the same parametervalues, but only for three regimes, the laissez-faire equilibrium, the Standardized Approach withboth risk weights equaling one, ci = 1, and the VaR approach with confidence level p = 1.0%. Bytheir numerical analysis (ibid., pp. 150-158), they can draw the same main conclusions.

2Cf. Crouhy/Galai/Mark (2000, p. 68), Gordy/Lutkebohmert (2007, p. 15), Hull (2003, p. 626).In periods of stress, such as the 1990/91-recession in the US, one-year default probabilities of BB-rated loans can even rise up to 3.5% whereas those of BBB-rated loans keep around 0.6%, asdo Ervin/Wilde (2001 p. S30) report. Following Carey (1998), Furfine (2001, p. 52) reports thatroughly one third of total loans can be asserted each to the rating classes BBB and BB, respectively,in typical bank loan portfolios, and Illing/Paulin (2005, p. 169) state that, concerning Canadianbanks, “a little more than two-thirds of bank exposures [are] rated investement grade (i.e., BBB-or higher)” , thus underpinning the empirical relevance of the credit risk structure assumed by the

4.2. EQUITY SHOCKS 119

risk weight of 1.0 according to Table 3.4. But we will also consider a risk

weight of c1 = 0.5 to account for the lower risk of Firm 1. The correlation,

Corr[L1/L1, L2/L2] ≈ 0.28, is, interpreted as a loan-return correlation, relatively

high. The Basel Committee assumes the average asset-return correlation to

take values from [0.12, 0.24] in the formulæ for the IRB-risk weights (BCBS,

2004, Art. 272). Moreover, asset-return correlations always translate to loan-

return correlations that are much lower, unless they approach one, as shown by

Gersbach/Lipponer (2003, p. 365f). In all comparative static analyses considered

in the next sections, the household’s and subsequently the bank’s decisions are not

constrained by the household’s initial wealth WH .

4.2 Equity Shocks

4.2.1 Total Loan Volumes and Pro-Cyclical Effects

Figure 4.1 shows the results of a comparative static analysis with respect to changes

in the bank’s initial equity WB. Both forms of regulation amplify the sensitivity

of the total loan volume to changes in WB. Thus, according to Definition (2.2.1),

regulation is pro-cyclical. Let us separate the single building blocks that add up to

the total effect.

First, in WB = 0, the bank issues a strictly positive amount of deposits and provides

loans in the absence of regulation according to Result 5, while it is constrained

under the VaR approach and its lending activities are scaled down to zero under

the Standardized Approach. Second, when WB is high enough, both Regulatory

Constraints (3.3.21) and (3.3.45) ease and are no longer binding and the total loan

volume is the same under all regimes. Third, under a binding regulatory constraint,

the total regulated loan volume is always lower than the non-regulated total loan

volume. Fourth, the total loan volumes strictly increase in WB under all three

regimes. The fact that the unregulated loan volume is the least sensitive toward

changes in WB, finally guarantees the pro-cyclical effect. In numbers, the equal

weighting under the Standardized Approach leads to a slope of 12.5, whereas the

total loan volume LV increases by 2.95 on average under the VaR approach with a

confidence level of 1− p1 = 0.5% and by only 1.28 without regulation.

Recall that the total loan/deposit volume is positive in WB = 0 under the VaR

parameter values given in Table 4.1. Similar numbers can be found in Gordy (1998) which arebased on data from internal Federal Reserve Board surveys.


0

002

004

006

008

0001

0021

0041

0061

0081

082062042022002081061041021001080604020ytiuqelaitinis'knaB

Tota

l loa

n vo

lum

enoitalugero/w

dezidradnatshtiwhcaorppa

1=2c=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%5.0

Figure 4.1: Total loan volume as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium total loan volume L = L1 + L2 on thebank’s initial equity WB. The total loan volume is shown for four different regimes: the laissez-faire equilibrium, the Standardized Approach with both risk weights equaling one, ci = 1, the VaRapproach with confidence levels p = 1.0% and p = 0.5%. Cf. Buhler/Koziol/Sygusch (2008, Fig.1, p. 152).

approach to regulation, since the bank can choose the loan-allocation rate ` such

that the regulatory constraint function v(`, 1 − p2) becomes negative while still

attaining a level of confidence of 1 − p2 = 1%. Particularly, the bank chooses the

optimal loan-allocation rate and the optimal Case according to (3.3.48), resulting

in `V ≈ 1.0378% and jV = 2. From WB ≈ 25 on, it becomes more profitable for

the bank to choose a loan portfolio that satisfies the stronger confidence level of

1 − p1 = 0.5%, although it is not obliged to do so. Then the regulatory constraint

function becomes

v(`, 0.005) = p2α2(1− `)− (1− p1)α1`

while the bank chooses loan-allocation rates above α2

α1+α2as presented in Figure

4.3. The change in the confidence level, and hence in the structure of the regulatory

constraint function v(`, p), results in a jump of the total loan volume/deposit volume

at WB ≈ 25. Despite the rather extreme loan-allocation rates chosen around WB =

25, the loan allocation is compatible with Case 1 from WB ≈ 20 on.

Because the equilibrium deposit volume under the VaR approach will always be


0

002

004

006

008

0001

0021

0041

0061

0081


Tota

l loa

n vo

lum

enoitalugero/w


dna5.0=1c0.1=2c

ehttaRaVlevel%2.0


This figure illustrates the dependence of the equilibrium total loan volume L = L1 + L2 on thebank’s initial equity WB. The total loan volume is shown for three different regimes: the laissez-faire equilibrium, the Standardized Approach with different risk weights, where c1 = 0.5 and c2 = 1,and the VaR approach with confidence level p = 0.2%.

positive in WB = 0 for intermediate levels of confidence, here p ∈ 1 − p1, 1 − p2are chosen, and because zero equity always implies a zero amount of risky loans

under the Standardized Approach, the VaR approach tends to be less pro-cyclical

than the latter approach. However, according to Result 23, DV = 0 holds for the

tightest confidence levels; that is, for p < 1−p1. Then, both curves representing the

equilibrium volumes of total loans under either VaR regimes of regulation start at

zero and their slopes depend on the specific values that are attached to given levels

of risk.

The reactions of the loan volumes under the VaR approaches would be further

dampened by lower values of τ . Moreover, the domain on which regulation is binding

would be enlarged proportionately.

Figure 4.2 compares the total loan volume LV for p = 1− p1− p2 + q = 0.2% to the

unregulated volume and the volume under the Standardized Approach with differing

risk weights ci. In this example, LV is less pro-cyclical than LS, regardless of whether

the loans are weighted equally (cf. Fig. 4.1), or not. As the risk weight c1 has been

reduced from 1 to 0.5 (whereas c2 remains fixed to 1), the regulatory constraint

function k(`) increases (for negligible indirect effects via `S). Thus the slope of the


%0

%01

%02

%03

%04

%05

%06

%07

%08

%09

%001


Loan

allo

catio

n ra

tenoitalugero/w

dezidradnatshcaorppa

ehttaRaVlevel%1

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.3: Loan-allocation rate as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium loan-allocation rate ` on the bank’sinitial equity WB. The loan-allocation rate is shown for four different regimes: the laissez-faireequilibrium, the Standardized Approach with both risk weights equaling one, ci = 1, the VaRapproach with confidence levels p = 1.0% and p = 0.5%. Cf. Buhler/Koziol/Sygusch (2008, Fig.2, p. 153).

feasible total loan volume becomes steeper with respect to WB. This reduction in

the risk weights affects the total loan volume LS in a pro-cyclical manner compared

to both the unregulated case and the case of equal risk weights.

The loan allocation slightly affects the total loan volume, as shown by Figure 4.2 for

the Standardized Approach with different risk weights. It is a only piecewise straight

line. The different slopes of the total loan volume in the Standardized Approach,

as shown in Figure 4.2 can be traced back to the equilibrium loan-allocation rates

which are depicted as a function of the bank’s equity in Figure 4.4: the total loan

volume LS increases by a slope of 25 for WB ≤ 13 because the loan-allocation rate

remains at one, `S = 1.00. From WB = 23 on, LS strictly increases at an almost

constant rate of roughly 16.15, whereas the total loan volume fluctuates in between:

first, `S falls from one to 0.892 for WB ∈ [13, 21], and second, `S drops from 0.892

to 0.817 at WB = 22 as the loan allocation migrates from Case 3 to 1.

Concerning the Standardized Approach, we can conclude that regulation seems to

be more pro-cyclical as the eligible risk weights decrease.


%0

%01

%02

%03

%04

%05

%06

%07

%08

%09

%001


Loan

allo

catio

n ra

tenoitalugero/w


dna5.0=1c0.1=2c

ehttaRaVlevel%2.0

Figure 4.4: Loan-allocation rate as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium loan-allocation rate ` on the bank’sinitial equity WB. The loan-allocation rate is shown for three different regimes: the laissez-faireequilibrium, the Standardized Approach with different risk weights, where c1 = 0.5 and c2 = 1, andthe VaR approach with confidence level p = 0.2%.

4.2.2 Tighter Confidence Levels and Total Volumes

If the level of confidence is tightened to 1 − p1 = 0.5%, the bank does exactly

comply with this level as long as regulation is effectively binding. In contrast, the

bank voluntarily switches to this stricter confidence level at WB ≈ 25 when the

regulatory confidence level is set equal to 1− p2 = 1%, as illustrated in Figures 4.1

and 4.3.

There are two off-setting effects that give rise to this switch from the weak to the

tight level of confidence: granting a higher loan volume to Firm 2 than to Firm

1 is favored if the bank must comply with the weak level, i.e. with p = 1 − p2.

As a result, there are higher expected gains on the loan portfolio which must be

rewarded by a higher deposit interest rate to the household, however. If the bank

voluntarily decides to meet the tighter confidence level p = 1−p1, expected gains on

the loan portfolio will be lower as a preference for Loan 1 will relax the regulatory

constraint. As this portfolio composition is less risky, the deposit interest rate is

also lower. Furthermore, the bank may even collect a higher deposit volume.

In this example, the bank chooses its optimal loan-allocation rate and its optimal


deposit interest rate according to Case 2 if p = 1− p2 is required, and according to

Case 3 if p = 1− p1 must be met, as long as WB ≤ 16 holds. In particular, the bank

allocates its funds according to (3.3.48) and (3.3.49), respectively at WB = 0. The

reason is that the regulatory function vanishes at WB = 0, i.e. v(`V , p) + ε = 0, and

the bank can freely scale the deposit volume Du(`V , RD; jV ) via RD.

For WB ≤ 25, the two offsetting effects are in numbers as follows: the difference

in the expected gross return on deposits due, qjRV,1−p2D − qjR

V,1−p1D , ranges from

2.0% up to 2.5%, while the difference in expected gross returns on the loan portfolio

ranges between 3.9% and 4.2%. At WB = 25, the marginal expected return on

intermediation is still by 1.9 percentage points larger for the bank if it mets p = 1−p2

than if it complies with a confidence level equal to p = 1−p1. But under the tighter

confidence level the bank can collect by far more deposits (the difference is equal

to approximately 270 dollars), despite of the lower interest rate on deposits to be

promised, such that the bank is in total slightly better off by following p = 1− p1.

To sum up, the bank may voluntarily obey to tighter confidence levels than it is

obliged to do and a tighter VaR-based regulation does not always result in lower

total loan/deposit volumes. Figure 4.1 illustrates this issue by the jump in the total

loan volume at WB = 25 if p = 1− p2 is required.

4.2.3 Single Loans and Pro-Cyclical Effects

Single loan volumes Li may react to changes in the bank’s initial equity differently

than the respective total loan volume L = L1 +L2. Different ways of regulation also

cause differences in sensitivities.

Under the Standardized Approach with equal risk weights the loan volume of Firm

1 remains zero for WB ≤ 24, as Figures 4.3 and 4.5 show. At the point where LS1

turns positive, the equilibrium supply of Loan 2 exhibits a kink (cf. Fig. 4.6). At

approximately WB ≈ 30, the equilibrium migrates from Case 2 to Case 1 which is

reflected by the second kink in the loan supply to Firm 2 because the fraction of

total loans that can be devoted to Firm 2 must be reduced considerably under Case

1 compared to Case 2.

The low total loan volumes LS forWB close to zero under the Standardized Approach

with equal risk weights are accompanied by loan-allocation rates `S equal or close

to zero. Obviously, the bank thus seeks to set off low volumes with high returns.

Figure 4.3 shows that this behavior basically holds for higher equity values WB as

well, albeit to a lesser extent.


0

001

002

003

004

005

006

007

008

009

0001


Loan

1noitalugero/w

ehttaRaVlevel%5.0

1=2c=1c


ehttaRaVlevel%0.1

Figure 4.5: Loan to Firm 1 as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium loan volume granted to Firm 1 on thebank’s initial equity WB. The loan volume L1 is shown for four different regimes: the laissez-faire equilibrium, the Standardized Approach with both risk weights equaling one, ci = 1, the VaRapproach with confidence levels p = 1.0% and p = 0.5%.

Under the VaR approach, single loan volumes can be larger than in the case of an

unregulated bank though the total loan volume is smaller, cf. Figures 4.5 to 4.7.

The reason is twofold: first, the bank chooses rather extreme loan-allocation rates

for intermediate levels of confidence and low values of equity. Figure 4.3 illustrates

that point. Particularly, a confidence level of p = 1 − p2 favors Firm 2 whereas

p = 1− p1 makes the bank prefer to grant credit for Project 1. If, additionally, the

regulated total loan volumes are closer to the unregulated total loan volumes than

they are to zero, the volume of a single loan under the VaR regulation may be higher

than the volume of its unregulated counterpart. With both the regulated and the

unregulated total loan volume close to each other, a single loan volume may exceed

its unregulated counterpart even if the level of confidence is set to p = 1−p1−p2 +q.

Figure 4.7 illustrates that point for LV2 and L∗2 if p = 1−p1−p2 +q = 0.2%. Figures

4.5 to 4.7 show the reactions of single loan volumes toward equity shocks.

Therefore a single firm may even benefit from risk-based capital requirements even

though it undertakes more risky projects than its competitors. This effect is also

found in the following two numerical examples analyzed in Sections 4.3.1 and 4.4.1.

As regulated loan-allocation rates do not move parallel to their unregulated


0

001

002

003

004

005

006

007

008


Loan

2

noitalugero/w

dezidradnats

ehttaRaVlevel%1

ehttaRaVlevel%5.0

1=2c=1chtiwhcaorppa


This figure illustrates the dependence of the equilibrium loan volume granted to Firm 1 on thebank’s initial equity WB. The loan volume L1 is shown for four different regimes: the laissez-faire equilibrium, the Standardized Approach with both risk weights equaling one, ci = 1, the VaRapproach with confidence levels p = 1.0% and p = 0.5%. Cf. Buhler/Koziol/Sygusch (2008, Fig.3, p. 154).

counterparts, single loan volumes granted under regulation do not always show pro-

cyclical patterns even if total loan volumes do. Particularly, LS1 is not pro-cyclical

for WB ≤ 24 when the risk weights are equal, i.e. c1 = c2 = 1, implying `S = 0

(Fig. 4.5). Non-pro-cyclical behavior can also be observed for LS2 if risk weights

distinguish between credit risk, c1 = 0.5, c2 = 1, and the bank’s initial equity is

small, i.e. WB ≤ 13, resulting in `S = 1. (cf. Fig. 4.7). Conversely, LS2 evolves

pro-cyclically for c1 = c2 = 1 and LS1 does so for c1 < c2 except for the areas where

migrations from one Case to another occur.

Under the VaR approach things are even more diverse: the loan granted to Firm 1,

LV1 , is counter-cyclical for p = 1− p1 = 0.5% and pro-cyclical for p = 1− p2 = 1.0%

and WB < 25. For p = 1 − p1 = 0.5%, LV2 is pro-cyclical for all WB and not

pro-cyclical if p = 1− p2 = 1.0% and WB < 25.3

Also the loan volume granted to Firm 2, LV2 , under p = 1−p1−p2 +q = 0.2% shows

pro- and non-pro-cyclical behavior, albeit LV2 is very flat from WB = 210 on.

To conclude, there seems to be a tendency that single loan volumes are pro-cyclical

3The respective slopes are approximately ∆LV2

∆WB∈ (0.48, 0.58) vs. ∆L∗2

∆WB≈ 0.73.


0

001

002

003

004

005

006

007

008


Loan

2noitalugero/w


dna5.0=1c0.1=2c

ehttaRaVlevel%2.0


This figure illustrates the dependence of the equilibrium loan volume granted to Firm 1 on thebank’s initial equity WB. The loan volume L1 is shown for three different regimes: the laissez-faireequilibrium, the Standardized Approach with different risk weights, where c1 = 0.5 and c2 = 1, andthe VaR approach with confidence level p = 0.2%.

if their volumes under regulation are lower than their unregulated equivalents.

4.2.4 Return Volatilities of Total Loans and Deposits

Without regulation, the risk-neutral bank faces a risk-return trade-off that is

solely caused by the risk-averse household’s deposit supply. The equilibrium loan-

allocation rate is characterized by this difference in risk attitudes. It is shifted in

favor of the high-return loan from that loan-allocation rate that would be chosen

by the household. This disagreement vanishes if the firms’ projects are identically

distributed. In this section, we discuss how this conflict of interest is resolved under

the different regimes.

Figures 4.3 and 4.4 show the dependence of the loan-allocation rate ` on the bank’s

initial equity WB under different regimes. Without regulation, the decrease in `∗

with increasing WB is based on two reasons: on the one hand, increasing bank

capital generally means that the monopolistic impact of the risk-neutral bank-owners

increases as there is a maximum deposit supply characterized by RD(`; j). As a

consequence, `∗ moves in favor of the bank owners. The higher the value of WB,


060.0

070.0

080.0

090.0

001.0

011.0

021.0


Ret

urn

vola

tiliti

es

noitalugero/w)stisoped(

hcaorppa

)stisoped(

ehttaRaVlevel%1)stisoped(

noitalugero/w)snaol(

hcaorppa

)snaol(

ehttaRaVlevel%1

)snaol(

1=2c=1c

1=2c=1c

dezidradnats

dezidradnats

Figure 4.8: Return volatilities as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium return volatilities on the bank’s initialequity WB. Solid and dashed lines represent volatilities of returns on deposits, σD. Diamondsand crosses represent volatilities on total loans, σ. Return volatilities are shown for three differentregimes: the laissez-faire equilibrium, the Standardized Approach with both risk weights equalingci = 1, and the VaR approach with confidence level p = 1%. Source: Buhler/Koziol/Sygusch (2008,Fig. 4, p. 155).

the greater the share of equity to total funds and the higher the fraction 1− `∗ the

bank grants as a loan to the second, the high-risk (and high-return) firm. On the

other hand, higher bank capital means a larger cushion for depositors. The higher

the value of WB, the higher the recovery rate of deposits so that deposit supply

increases other things being equal, as it has been pointed out for some specific

equilibria in Chapter 3. Hence, the bank may lower the loan-allocation rate `∗ and

thus increase the expected returns and volatilities without forgoing an increase in

deposit volume or without increasing deposit interest rates. Figure 4.2 illustrates

this point in connection with Figures 4.3 and 4.10. Concerning the volatility of

return on total loans, we refer to Figure 4.8.

Figure 4.3 shows that regulation by identical risk weights has adverse effects on

the optimal portfolio. The low-risk loan is always less weighted than it is without

regulation. This choice by the bank owners can be explained as follows: under

the Standardized Approach with equal risk weights, the total loan volume that the

bank can grant is fixed in proportion to WB. Hence, with decreasing equity, the

total loan volume shrinks by a rate of 12.5. The bank offsets the low volume with


060.0

560.0

070.0

570.0

080.0

580.0

090.0

590.0

001.0


Ret

urn

vola

tiliti

es



)stisoped(

ehttaRaVlevel%2.0

)stisoped(



)snaol(

ehttaRaVlevel%2.0

)snaol(

1=2c,5.0=1c

1=2c,5.0=1c


This figure illustrates the dependence of the equilibrium return volatilities on the bank’s initial equityWB. Solid and dashed lines represent the volatilities of returns on deposits. Diamonds and crossesrepresent the volatilities of returns on total loans. Return volatilities are shown for three differentregimes: the laissez-faire equilibrium, the Standardized Approach with different risk weights, wherec1 = 0.5 and c2 = 1, and the VaR approach with confidence level p = 0.2%.

higher expected returns by widening the fraction of total capital granted as a loan

to the second firm.

If risk weights sufficiently account for differences in the credit risk of loans, the

regulated bank may prefer the low-risk loan over the high-risk loan. Unlike when

it is dealing with equal risk weights, the bank chooses a loan-allocation rate of

one, `S = 1.00, if WB ≤ 13 holds given c1 = 0.5 and c2 = 1. For WB ≥ 14

until regulation ceases to bind, `S > `∗ still holds (cf. Figures 4.3 and 4.4). But

as a consequence, regulation mainly corrodes incentives for diversification. Let us

measure the depositor’s risk by the return volatility σD of deposits, defined as

σD ≡√

V(DjD

) , (4.2.1)

where j = 1, . . . , 4 denotes one of the Cases and Dj is the stochastic, Case-dependent

and state-dependent redemption of deposits as partially outlined in (3.2.23) and

(3.2.23).

Figures 4.8 and 4.9 show the return volatilities of the bank’s loan portfolios and

the return volatilities of the associated deposits under different regimes. The former


are depicted by diamonds for the unregulated case and for the VaR approach and

by crosses for the Standardized Approach. The latter are represented by solid and

dashed lines, respectively.

Clearly, the bank absorbs the return volatility from its loan portfolio such that the

returns on deposits are always less volatile. Two mechanisms are at work: the pay-off

structure of the deposit contract truncates extreme (though favorable) outcomes and

thus lessens volatility, whereas the residual volatility is absorbed by the bank’s initial

equity WB. These two effects can be particularly well studied in the case without

regulation: The gap between the volatilities of returns on deposits (black solid line)

and the volatilities of returns on total loans (black diamonds) at WB = 0 reflects the

cut in volatility by the upper bound the deposit contract imposes, while the growing

gap between those volatilities represents the cushion effect. The risk-absorbing role

banks play is emphasized by Bitz (2006, p. 4, pp. 14-21). Greenbaum/Thakor (2007,

p. 108) understand this risk transformation function, which is enhanced by the

property that single loan volumes often exceed single deposit volumes, even as a

driver for monopoly power in banking.

Figure 4.9 shows that depositors are not better protected for all levels of bank

equity by the Standardized Approach than they are in case of an unregulated bank.

In fact, the depositor is worse off in these terms under the Standardized Approach

whenever the return volatility on the regulated loan portfolio is higher than the

return volatility on the unregulated loan portfolio. One observes a similar result

if risk weights are fixed, cf. Figure 4.8: depositors face a lower return volatility on

their deposits under regulation compared to the unregulated case if the regulated

loan portfolio’s return volatility approaches that of the unregulated loan portfolio.

Lowering the risk weight c1 from 1 to 0.5 reduces the overall return volatilities for

both the loan portfolio and for deposits, as the comparison of Figure 4.8 and Figure

4.9 reveals.

This also applies to the volatilities of the returns on deposits under the VaR

approach: a stricter level of confidence decreases the deposit return volatility.

However, this is not true for the volatilities of portfolio returns. Furthermore,

depositors may bear more risk in terms of volatilities under the VaR approach

than they do under the unregulated regime if the high-risk loan LV2 outweighs its

unregulated counterpart L∗2 in volume. By and large, the deposit return volatilities

under p = 1− p1 = 0.5% are between the return volatilities under the next weaker,

1− p2, and the next stricter, 1− p1 − p2 + q, level of confidence.

Figures 4.8 and 4.9 show, in essence, that the stricter confidence levels are under a


VaR approach or the more risk weights distinguish between high- and low risk loans,

the better the protection for the depositors. Despite this property, the depositors

may still bear more risk in terms of return volatilities than they do in a world

without regulation. This holds regardless of whether regulation is risk-sensitive or

not. In contrast, the volatility of deposit redemptions, V (D), is always reduced

through regulation which is due to the volume reductions.

4.2.5 Deposit Interest Rates

Figure 4.10 shows that the deposit interest rate R∗D offered by the unregulated bank

strictly decreases in WB, reflecting the fact that as WB increase, it becomes easier

for the bank to collect deposits from the risk-averse household.

The behavior of the deposit interest rate under the various regulatory regimes can be

essentially traced back to two roots: increasing interest rates are present if the bank

primarily seeks to collect more deposits as the maximum feasible total loan/deposit

volume increases with WB. We refer to this phenomenon as the volume aspect. In

this context, the bank suffers from the positive relation between deposit volume and

deposit interest rates in equilibrium, as stated in Result 2.

The risk aspect is understood to be the property that the deposit interest rate

decreases with increasing equity, as the default risk of deposits thus decreases.

In particular, the volume aspect outweighs the risk aspect if the risk changes only

slightly or if it remains unchanged, as is the case under the Standardized Approach

for low values of bank equity. Interest rates strictly decrease with WB under

regulation if the bank considerably reduces risks with increasing equity. This is

particularly true under the VaR approaches for low values of equity. Otherwise, the

volume effect dominates.

Although the deposit interest rate seems to be the main instrument for the bank

to reduce deposit volumes to the volumes that are allowed by regulation, regulation

may induce the bank to choose optimal loan portfolios that are so risky that the bank

must compensate the risk-averse depositor for the higher risk with higher interest

rates. Thus, the regulated bank promises higher deposit interest rates than it would

do without regulation. Such an increase in deposit interest rates can be observed

under the VaR approach for p = 1− p2 = 1% and WB < 25, as illustrated in Figure

4.10.

In contrast, deposit interest rates under regulation are always lower than their


050.0

060.0

070.0

080.0

090.0

001.0

011.0

021.0

031.0

041.0


Dep

osit

inte

rest

rate

noitalugero/w


1=2c=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%5.0


dna5.0=1c0.1=2c

ehttaRaVlevel%2.0

Figure 4.10: Deposit interest rate as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium deposit interest rate RD on the bank’sinitial equity WB. Deposit interest rates are shown for all six different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with the confidence levels p = 1%, p = 0.5% and p = 0.2%.

unregulated counterparts for equally distributed firm projects because the minimum

variance portfolio is chosen regardless of the regime; i.e. ` = 12

holds, as shown by

Results 15, 22, 26, and 29.

Concerning cyclicality, we can observe the following: if the risk aspect seems to

dominate the choice of the deposit interest rate, deposit interest rates may become

pro-cyclical since they decrease under both regimes. If the volume aspect seems

to dominate the choice of the deposit interest rate, deposit interest rates become

counter-cyclical since they increase, while R∗D strictly decreases.

4.3 Credit Risk Shocks

4.3.1 Total Volumes and Cyclical Dampening

Shocks in credit risk are modeled by parallel shifts in the single and in the common

success probabilities. That is, the parameters p1, p2, and q are scaled by a common

multiplier m. The lower Bound (3.2.11) on the joint default probability q implies

1− p1 − p2 + q > 0, and thus, if all the probabilities p1, p2, and q are jointly scaled

4.3. CREDIT RISK SHOCKS 133

0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062

0082

200.1100.1000.1999.0899.0799.0699.0599.0499.0399.0299.0199.0seitilibaborpsseccusehtrofreilpitlumnommoC

Tota

l loa

n vo

lum

e

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%5.0

ehttaRaVlevel%2.0

htiwhcaorppa

htiwhcaorppa

Figure 4.11: Total loan volume as a function of shifts in success probabilities

This figure illustrates the dependence of the equilibrium total loan volume L on shifts in eachproject’s success probability pi and in their common success probability q. Shifts in successprobabilities are modeled using a common multiple m, 0.991 ≤ m ≤ 1.002. Total loan volumesare shown for five different regimes: the laissez-faire equilibrium, the Standardized Approach withboth equal and different risk weights, and the VaR approach with the confidence levels p = 0.5%and p = 0.2%.

by m, results in the following upper bound for m whereas Condition (3.2.6) in the

following lower bound such that m is restricted to

Rf

p1α1

< m <1

p1 + p2 − qDef. (3.2.41)≡ 1

q4

.

Economically, the lower bound is set such that the expected gross return on the

first firm’s project exceeds the risk-free rate. The upper bound guarantees that the

deposit redemption remains risky under Case 4. Furthermore, the upper bound

implies mpi < 1, as pi < q4, i = 1, 2.

As a result, the correlation Corr(X1, X2) strictly decreases in m,

∂Corr(X1, X2)

∂m= −p1(p2 − q)(1−mp2) + p2(p1 − q)(1−mp1)

2√p1p2 ·

√(1−mp1)(1−mp2)

3 < 0 . (4.3.1)

Given the bounds imposed on m from above, the correlation between the projects


0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062

0082

200.1100.1000.1999.0899.0799.0699.0599.0499.0399.0299.0199.0seitilibaborpsseccusehtrofreilpitlumnommoC

Tota

l loa

n vo

lum

e

%1 level

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%5.0

ehttaRaV

htiwhcaorppa

htiwhcaorppa

Figure 4.12: Total loan volume as a function of shifts in success probabilities

This figure illustrates the dependence of the equilibrium total loan volume L on shifts in eachproject’s success probability pi and in their common success probability q. Shifts in successprobabilities are modeled using a common multiple m, 0.991 ≤ m ≤ 1.002. Total loan volumesare shown for five different regimes: the laissez-faire equilibrium, the Standardized Approach withboth equal and different risk weights, and the VaR approach with the confidence levels p = 1% andp = 0.5%. Cf. Buhler/Koziol/Sygusch (2008, Fig. 6, p. 157).

thus varies between

q(p1 + p2 − q)− p1p2√p1p2(p1 − q)(p2 − q)

< Corr(X1, X2) <qα1 − p2Rf√

p2(α1 −Rf )(p1α1 − p2Rf ).

Given the parameter values from Table 4.1, we let the multiplier m take values from

[0.9500, 1.0020] in steps of 0.0001. The upper bound is slightly below 1p1+p2−q . The

lower bound was chosen because the regulation under fixed risk weights becomes

binding for values far above 0.9500 (cf. Fig. 4.11), and because the VaR approach

with a confidence level of p = 1% does not result in strictly positive deposit volumes

for m < 0.9920 (cf. Fig. 4.12).4

Using the parameter values from Table 4.1, the correlation lies in the interval

(−0.004936, 0.9383). The parameter choice m ∈ [0.9500, 1.0020] reduces this range

to [−0.004122, 0.9039] (bounds rounded to four leading digits each).

Because lower values for the multiplier m imply higher probabilities of default, low

values of m can be associated with a less favorable state of the economy. Moreover,

4The total loan volume LV reduces to the bank’s initial equity WB , in this example WB = 100.


%6

%7

%8

%9

%01

%11

%21

%31

%41

%6.98%6.97%6.96%6.95%6.94%6.93%6.92%6.91%6.9%4.0

noitalerroC

Dep

osit

inte

rest

rate

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%1

ehttaRaVlevel%5.0

ehttaRaVlevel%2.0

htiwhcaorppa

htiwhcaorppa

Figure 4.13: Deposit interest rate as a function of the correlation

This figure illustrates the dependence of the equilibrium deposit interest rate RD on the projectreturns’ correlation. Changes in the correlation are based on joint shifts in each project’s successprobability pi and in their common success probability q. These shifts are modeled using a commonmultiple m, 0.991 ≤ m ≤ 1.002. Deposit interest rates are shown for all six different regimes: thelaissez-faire equilibrium, the Standardized Approach with both equal and different risk weights, andthe VaR approach with the confidence levels p = 1%, p = 0.5% and p = 0.2%.

a bad economic situation in terms of higher default probabilities is linked to a

higher (default) correlation, since Corr(X1, X2) strictly decreases in m according to

(4.3.1).5 The regulatory formula for the average asset-return correlation establishes

the qualitative relation (cf. BCBS, 2004, Art. 272) which has been confirmed

by empirical studies such as those by Lopez (2004) and Vondra/Weiser (2006).

Beyond empirical evidence and the purely mathematical relation between default

probabilities and default correlations, covariations in asset prices may even arise in

downturns when asset prices are otherwise stochastically independent. This may

be due to agents’ wealth constraints or fire sales, in particular due to risk-sensitive

capital regulation as Danıelsson/Zigrand (2008) show in their general equilibrium

model.

Figures 4.11 and 4.12 show the total loan volumes under the various regimes

dependent on shifts in the success probabilities p1, p2, and q by the multiplier m,

5Erlenmaier/Gersbach (2001) use a Merton-type framework and show in their Proposition 1,p. 6, that default correlations increase for increasing default probabilities as long as they remainbelow 0.5. Gersbach/Lipponer (2003) take a broader view on the relation between shocks, defaultprobabilities, and default correlations.


with m = 1 referring to the base case. The higher these success probabilities are,

the higher the total loan volume without regulation becomes. This is intuitive:

the lower the default risk, the higher the propensity of the risk-averse household

to supply deposits for given loan-allocation rates and for given deposit interest

rates. Moreover, as risks are reduced, the bank can even lower deposit interest

rates in equilibrium, as illustrated in Figure 4.13. The abscissa is drawn in terms of

correlations and thus the graphs must be reversed in terms of the multiplier m.

Especially, deposit interest rates increase under all regimes with the loans’

correlation, hence with aggregate risk. This monotonic relation holds except for

the point at which the equilibrium changes from one Case to another, and the point

at which the structure of the level of confidence effectively attained by the bank

changes.

The change from Case 1 to Case 2 happens at a correlation of 59.89% (m = 0.9940,

cf. also Fig. 4.11 and 4.14) in the laissez-faire equilibrium. The steep ascent in R∗D

thereafter reflects a reduction in the loan-allocation rate from `∗ = 25.59% to zero,

resulting in higher risks for the depositor beyond the shifts in the correlation/the

probabilities.

If a confidence level of p = 0.5% is required, the bank will decide to comply

with p = 1 − mp1 − mp2 + mq from a correlation of 27.79% on, i.e. below

m = 1.0000. As a consequence, the bank effectively attains confidence levels between

0.2% and 0.4994% for correlations ranging from 28.76% to 48.56% (corresponding

to multipliers from m = 0.9999 down to 0.9970). The change is also reflected in

a jump in the total loan volume granted, as shown in Figure 4.11 with respect to

m and in Figure 4.14 with respect to the correlation. From m = 1.0000 on (that

is, for correlations equal to and below 27.79%) the bank attains confidence levels

varying from 0.3010% to 0.5%, corresponding to the structural level of 1−mp1. The

associated higher deposit interest rate results in a sudden increase of the total loan

volume (cf. Fig. 4.11 and 4.14).

If the bank must comply with p = 1.0%, there is another jump from m = 0.9926 to

0.9927. This jump is solely characterized by a considerable increase in the loan-

allocation rate from 27.06% to 33.84%. At this point, there is neither a Case

migration nor a structural change in the level of confidence that is effectively met.

But a structural change such as this also occurs under this regulatory regime, namely

at m = 0.9960, corresponding to a correlation of 53.01%. Below this critical value

of m, the bank effectively complies with a structural level of 1−mp1 −mp2 + mq,

corresponding to confidence levels ranging from 0.61% to 1.00%. Above this critical


0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062

0082

%6.98%6.97%6.96%6.95%6.94%6.93%6.92%6.91%6.9%4.0noitalerroC

Tota

l loa

n vo

lum

e

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%5.0

ehttaRaVlevel%2.0

htiwhcaorppa

htiwhcaorppa

Figure 4.14: Total loan volume as a function of the correlation

This figure illustrates the dependence of the equilibrium total loan volume L on the project returns’correlation. Changes in the correlation are based on joint shifts in each project’s success probabilitypi and in their common success probability q. These shifts are modeled using a common multiple m,0.991 ≤ m ≤ 1.002. Deposit interest rates are shown for all six different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with the confidence levels p = 1%, p = 0.5% and p = 0.2%.

value of m, the bank voluntarily complies with a level equal to 1−mp1, corresponding

to values from 0.30% to 0.90%.

Deposit interest rates are smaller under regulation because the deposit volume is

lower and risks are reduced. However, they react more strongly on changes in

correlation and success probabilities. Thus, regulation affects interest rates in a

pro-cyclical manner even if it does not do so with the total loan/deposit volumes.

As the total loan volume is strictly increasing in m and as the total loan volume that

is feasible under regulation is always lower than the unregulated total loan volume,

there is a m above which regulation is binding. As regulation remains binding from

each of these critical values for m on, the regulated total loan volumes are on average

not pro-cyclical. Inversely, the regulated bank behaves as if it were unregulated in

severe economic contractions (equivalent to higher correlations or lower values for

m). Hence, regulation rather deters the bank from participating in the fruits of an

upswing. These issues are illustrated in Figures 4.11, 4.12, and 4.14.

There is, however, one exception in our example. Under the VaR approach with


Tab

le4.2:

Eff

ects

on

the

tota

llo

an

volu

me

by

VaR

-base

dre

gula

tion,

Bern

oulli

model

This

tablereports

equilibriaunder

aV

aR-based

regulationw

itha

requiredconfidence

levelof

1%and

thecorresponding

equilibriain

thelaissez-faire

economy.m

isthe

jointm

ultiplierreferring

tothe

successprobabilities

pi

andto

thecom

mon

successprobability

q.O

therwise,

allparam

etersare

givenas

inT

able4.1.

We

notethat

thetotal

loan/depositvolum

eunder

regulationexceeds

itsunregulated

counterpartfor

0.9927≤m≤

0.9940.

w/o

regulation

VaR

regulation

at1%

resultingin

resultingin

p∗

=1−

mp

2pV

=1−

mp

1 −mp

2+mq

for0.9862

≤m≤

0.9967

for0.9920

≤m≤

0.9959

mC

orr(X1 ,X

2 )` ∗

R∗D

j ∗D∗

E[W

∗B ]σ∗

`V

RVD

jV

DV

E[W

VB]

σV

0.992064.99%

25.41%1.1222

2394

.2141

.40.01772

26.89%

1.11781

366.4

141.20.01746

0.992264.54%

25.22%1.1220

2396

.8141

.90.01750

26.95%

1.11571

366.7

141.60.01718

0.992464.08%

25.03%1.1219

2404

.8142

.30.01728

27.00%

1.11471

366.9

142.00.01690

0.992663.61%

24.84%1.1217

2410

.3142

.80.01706

27.06%

1.11371

367.2

142.40.01661

0.992763.36%

24.75%1.1216

2413

.1143

.10.01695

33.84%

1.11741

415.8

142.70.01583

0.993062.62%

24.47%1.1214

2421

.7143

.90.01662

35.47%

1.11671

428.8

143.50.01527

0.993262.10%

24.28%1.1212

2427

.6144

.40.01640

36.55%

1.11621

437.8

144.00.01490

0.993461.57%

24.09%1.1210

2433

.6144

.90.01619

37.63%

1.11561

447.2

144.60.01454

0.993661.03%

23.90%1.1209

2439

.8145

.50.01597

38.71%

1.11511

456.8

145.10.01418

0.993860.47%

23.71%1.1207

2446

.2146

.10.01575

39.79%

1.11461

466.8

145.70.01382

0.994059.89%

23.52%1.1205

2452

.7146

.70.01553

40.86%

1.11411

477.1

146.40.01347

0.994159.60%

33.65%1.1189

1489

.6147

.00.01420

41.40%

1.11341

482.4

146.70.01330

0.994458.68%

34.97%1.1183

1507

.3148

.00.01368

43.01%

1.11311

498.8

147.70.01278


40.0

50.0

60.0

70.0

80.0

90.0

0200.16100.12100.18000.14000.10000.16999.02999.0

seitilibaborpsseccusehtrofreilpitlumnommoC

Ret

urn

vola

tiliti

es



1=2c=1c)stisoped(


1=2c,5.0=1c)stisoped(



)snaol(


1=2c,5.0=1c)snaol(

1=2c=1c

Figure 4.15: Return volatilities as a function of shifts in successprobabilities

This figure illustrates the dependence of the equilibrium return volatilities on shifts in each project’ssuccess probability pi and in their common success probability q. Shifts in success probabilities aremodeled using a common multiple m, 0.991 ≤ m ≤ 1.002. Solid and dashed lines and dots,respectively, represent volatilities of returns on deposits, σD. Diamonds and crosses representvolatilities on total loans, σ. Return volatilities are shown for three different regimes: the laissez-faire equilibrium, and the Standardized Approach with both equal and different risk weights.

p = 1%, we observe that the total loan volume LV exceeds the unregulated total

loan volume L∗. This is surprising as a binding Regulatory Constraint 3.3.45

suggests that the regulated total loan volume is always lower than its unregulated

counterpart. But the required level of confidence is essential, too.

In this example, the bank keeps a level of confidence of 1 −mp2 for 0.9927 ≤ m ≤0.9940 (corresponding to correlations ranging between 63.36% and 59.89%) without

regulation. This structural level of confidence is associated with a low fraction of

total funds that the bank grants as a loan to Firm 1 (cf. Fig. 4.16). The result

is associated with Case 2. But if the bank is required to meet a confidence level

of 1%, it chooses a structural level of confidence equal to 1 − mp1 − mp2 + mq

for 0.9920 ≤ m ≤ 0.9959, corresponding to correlation values between 64.99% and

53.41%.

The resulting reduction in risk by the VaR-approach is that essential for some

values of the multiplier m, that the regulated bank issues more deposits than if

it is unregulated. Table 4.2 illustrates this phenomenon in numbers. It can also


0

002

004

006

008

0001

0021

0041

0061

0081

%6.98%6.97%6.96%6.95%6.94%6.93%6.92%6.91%6.9%4.0noitalerroC

Loan

1noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaV

htiwhcaorppa

htiwhcaorppa

level%1

Figure 4.16: Loan to Firm 1 as a function of the correlation

This figure illustrates the dependence of the equilibrium loan volume granted to Firm 1 on theproject returns’ correlation. Changes in the correlation are based on joint shifts in each project’ssuccess probability pi and in their common success probability q. These shifts are modeled using acommon multiple m, 0.991 ≤ m ≤ 1.002. The loan volume L1 is shown for four different regimes:the laissez-faire equilibrium, the Standardized Approach with both equal and different risk weights,and the VaR with the confidence level p = 1%.

be seen in Figure 4.17. The regulated total loan volume exceeds its unregulated

counterpart where the absolute difference between the volatilities of gross deposit

returns is up to a multiple of 10 higher that it is the case close to this area. In

numbers, this gap ranges between 0.0012 and 0.00206 where the unregulated total

loan volume is higher and only between 0.00026 and 0.00090 where it is not.

From m = 0.9941 on, the unregulated total loan volume L∗ is again larger than LV

at the 1%-level. As long as LV > L∗ holds here, LV is also more sensitive toward

changes in WB. Hence, the VaR approach makes total lending pro-cyclical on this

domain.

Otherwise the VaR approach at the 1% confidence level is not pro-cyclical either, but

it shows the highest sensitivity except for the jump discontinuities with the other

VaR regimes (cf. Fig. 4.12). In contrast, the Standardized Approach with equal risk

weights has zero sensitivity to WB. The latter is obvious as by c1 = c2 the total

loan volume LS is given by 1c·c1 ·WB and hence LS is independent of changes in risk

unless c1 moves. Under the Standardized Approach with c1 = 0.5 and c2 = 1, the

total loan volume strictly increases in m albeit more slightly than L∗.


4.3.2 Single Loans and the Role of Constant Risk Weights

Figure 4.16 reveals that the bank, regulated with c1 = 0.5, pushes lending to Firm 1

when correlation decreases; that is, when success probabilities rise. Here again, the

lower risk weight makes the low-risk, but also low-return loan more attractive to the

bank as one dollar lent requires only 0.04 cents of equity, i.e. half the equity than it

is the case for c1 = 1. Thus, the total loan volume can be increased by increasing the

loan-allocation rate as the monotonicity and convexity of the regulatory constraint

function k(`) suggests, cf. (3.3.23) and (3.3.24). Higher loan volumes, in turn, lead to

higher expected wealth for the bank. In this example, the regulated bank increases

the total loan volume by choosing a loan-allocation rate `S that even exceeds `∗.

Figure 4.15 shows that the loan-allocation rate `S chosen under c1 = 0.5 reduces

the volatility of returns on the loan portfolio compared to the case with equal

risk weights and compared to the unregulated case. Comparing both instances

of the Standardized Approach shows that risk-reducing effect on the deposit return

volatility is fairly small and ambiguous in this example.

Although the loan-allocation rate `S is higher than `∗ for c1 = 0.5, the volume lent

to Firm 1 is not pro-cyclical according to Definition (2.2.2), neither with respect to

the correlation nor to the shift parameter m. Given c1 = c2 = 1, the volume granted

to Firm 1, LS1 , is counter-cyclical.

If c1 = 0.5 and c2 = 1.0, the high-risk firm, Firm 2, is granted less when expectations

become more optimistic. Concerning equal risk weights, we observe the opposite as

economies in equity are no longer present. That is, the better the prospects, the

higher the volume of LS2 becomes. The unregulated bank also increases L∗2 with

expectations turning better. As a result, LS2 is not pro-cyclical if c1 = c2 = 1 holds,

but is counter-cyclical for c1 = 0.5 with c2 = 1.

Although risk weights that reflect the differences in credit risk, in this example by

c1 = 0.5 and c2 = 1, result in loan portfolio compositions that are less risky than in

the unregulated case, equal risk weights also show a positive effect. If the success

probabilities decrease, the loan volume granted to the less risky loan increases,

albeit weakly. This stabilizing effect by regulation is, however, qualified by the fact

that regulation is no longer binding from a point on at which the probabilities are

sufficiently low.


4.3.3 Flexible Risk Weights

If we allow the risk weights ci to move with shifts in the default probabilities,

results will be mixed as risk weights will be both equal and different. Consider the

different eligible values of the risk weights according to the Standardized Approach,

cf. Table 3.4. Let ci = 0.5 if 1 − mpi ≤ 0.50% and ci = 1 if 1 − mpi ≤ 2.00%

is fulfilled, and ci = 1.5 else.6 Then the risk weights ci take the following values

dependent on m,

c1 =

1.5 if 0.9176 < m ≤ 0.9849

1.0 if 0.9850 ≤ m ≤ 1.0000

0.5 if 1.0001 ≤ m < 1.0021

, c2 =

1.5 if 0.9176 < m ≤ 0.9898

1.0 if 0.9899 ≤ m < 1.0021.

Comparing eligible total loan volumes to those chosen by the unregulated bank,

leads to the conclusion that regulation is binding on m ∈ [0.9995, 1.0000] and on

[1.0009, 1.0020]. Moreover, on [0.9995, 1.0000], both risk weights are equal to one,

whereas on [1.0009, 1.0020], c1 = 0.5 and c2 = 1 holds. In between, that is on

[1.0001, 1.0008], c1 = 0.5 and c2 = 1 holds true as well and regulation is not binding.

Thus, the results from the two different instances of the Standardized Aprroach

carry over to those two domains on which regulation is binding: the total lending

volume under binding regulation is not pro-cyclical. The volume lent to Firm 1 is

counter-cyclical on [0.9995, 1.0000] and not pro-cyclical on [1.0009, 1.0020]. From

a global perspective, total lending becomes erratic. Slight changes in the success

probabilities make the total loan volume migrate within three different regimes. The

switch at m = 1.0000 causes a jump in the total loan volume from 1, 250 to 1, 438.1.

As long as the regulatory constraint is not binding, the bank can widen its total

loan volume up to 1, 739.1. Thereafter, regulation is binding again and the growth

of the total loan volume becomes weaker, and the total loan volume ranges from

1, 789.8 to 1, 839.8 for 1.0009 ≤ m ≤ 1.0020.

4.3.4 Effects under the Value-at-Risk Approach

Apart from the jumps, the total loan volumes LV are not pro-cyclical compared to

L∗. Depending on the shifts in the success probabilities and the level of confidence,

regulation may even affect lending slightly counter-cyclically.

6The lowest feasible default probability in this example amounts to 1−mp1 > 1−1.0021·0.995 =0.0029105, still justifying a risk weight equal to 0.5.


First, let us consider the VaR approach with a level of confidence of 1%, as shown

in Figure 4.12. Because the success probabilities move with m, different structural

levels are eligible: below m = 0.9920, the regulated bank is not allowed to grant any

loans financed by debt. Hence, the regulated total loan volume equals the bank’s

initial equity, here 100. By the bank owners’ risk-neutrality, all capital is granted

as loan to Firm 2. For 0.9920 ≤ m ≤ 0.9950, the bank can only comply with a

confidence level of 1% by allocating its loans such that p = 1 − mp1 − mp2 + mq

is fulfilled. From m = 0.9950 on, the bank could already switch to a level of

p = 1 − mp1, which is not optimal, however, such that p = 1 − mp1 − mp2 + mq

is maintained till m = 0.9959. From m = 0.9960 on, the bank finally chooses p =

1−mp1. Although the bank owners could choose loan portfolios whose redemptions

conform to p = 1 − mp2 from m = 1.0000 on, they do not switch the structural

levels again. As a result, loan-allocation rates above α2

α1+α2are chosen.

Because the bank chooses `V ≤ α2

α1+α2for p = 1−mp1 −mp2 + mq, 0.9920 ≤ m ≤

0.9959 and `V > α2

α1+α2otherwise, the total deposit volume is tied to equity and

expected returns according to

DV =

(τ

m · p2α2(1− `V )− (1−m · p1)α1`V + ε− 1

)·WB

as stated in (3.3.46). DV is directly and indirectly, via `V , affected by m. The direct

effect,

∂DV

∂m= − τWB ·

[p2α2(1− `V ) + p1α1`

V]

[m · p2α2(1− `V )− (1−m · p1)α1`V + ε]2< 0 (4.3.2)

is always negative whereas the indirect effect,

∂DV

∂`· d`

V

dm= − τWB · [−m · p2α2 − (1−mp1)α1]

[m · p2α2(1− `V )− (1−m · p1)α1`V + ε]2· d`

V

dm(4.3.3)

is ambiguous as the behavior of the equilibrium loan-allocation rate is in general

unclear. DV strictly increases in ` because of p2α2 > p1α1.

For p = 1%, the loan-allocation rate `V strictly increases and the indirect effect

outweighs the direct effect resulting in a strictly increasing total loan volume LV as

soon as the bank is allowed to issue debt DV > 0 (which is the case from m = 0.9920

onwards). This increasing behavior is shown by Figure 4.12.

Issuing a strictly positive amount of deposits and compliance with a confidence level

of 0.5% is feasible from m = 0.9970 on (cf. Fig. 4.11 and 4.12), to one of 0.2%


0

002

004

006

008

0001

0021

0041

0061

0081

%6.98%6.97%6.96%6.95%6.94%6.93%6.92%6.91%6.9%4.0noitalerroC

Loan

1noitalugero/w

dezidradnats,hcaorppa

1=2c=1c


1=2c,5.0=1c

ehttaRaVlevel%5.0

ehttaRaVlevel%2.0

Figure 4.17: Loan to Firm 1 as a function of the correlation

This figure illustrates the dependence of the equilibrium loan volume granted to Firm 1 on theproject returns’ correlation. Changes in the correlation are based on joint shifts in each project’ssuccess probability pi and in their common success probability q. These shifts are modeled using acommon multiple m, 0.991 ≤ m ≤ 1.002. The loan volume L1 is shown for five different regimes:the laissez-faire equilibrium, the Standardized Approach with both equal and different risk weights,and the VaR approach with the confidence levels p = 0.5% and p = 0.2%.

from m = 1.0000 on (cf. Fig. 4.11). Under the 0.5% regime the bank can relax its

decisions to a level of 1 −mp1 from m = 1.0000 onwards which it effectively does.

Below it follows a structural level equal to 1 −mp1 −mp2 + mq, corresponding to

values between 0.2100% and 0.4994%. The migration from one structural level of

confidence to the other causes a jump at m = 1.0000 (correlation equal to 27.79%).

The bank’s behavior implies that the confidence levels of p = 1% and p = 0.5% yield

the same results for m = 1.0000 onwards (for correlations below the critical value of

27.79%). Especially, regulation does not have a pro-cyclical impact, neither on the

total nor on the single loan volumes on this domain.

But for m ≤ 0.9999, the VaR approach that requires a level of confidence of 0.5%,

affects total lending counter-cyclically: the optimal loan-allocation rate is equal toα2

α1+α2and thus only the direct effect is present. In absolute terms, the total loan

volume LV shrinks from 595.68 to 591.47. Compared to the slope of L∗, this counter-

cyclical effect is small: an absolute change in probabilities by 0.30% translates into

a relative cutback in the lending volume of −0.7%. These sensitivities have already

been discussed in connection with Result 24.


40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0

31.0

41.0

51.0

200.1100.1000.1999.0899.0799.0699.0599.0499.0399.0299.0199.0

seitilibaborpsseccusehtrofreilpitlumnommoC



1=2c=1c)stisoped(

%1ehttaRaV)stisoped(level


dezidradnatshcaorppa

)snaol(

%1ehttaRaV)snaol(level

1=2c=1c

Ret

urn

vola

tiliti

es

Figure 4.18: Return volatilities as a function of shifts in successprobabilities

This figure illustrates the dependence of the equilibrium return volatilities on shifts in each project’ssuccess probability pi and in their common success probability q. Shifts in success probabilitiesare modeled using a common multiple m, 0.991 ≤ m ≤ 1.002. Solid and dashed lines representvolatilities of returns on deposits, σD. Diamonds and crosses represent volatilities on total loans,σ. Return volatilities are shown for three different regimes: the laissez-faire equilibrium, theStandardized Approach with both risk weights equaling one, ci = 1, and the VaR approach withconfidence level p = 1%.

Under p = 0.2%, the bank always chooses `V = α2

α1+α2such that only the direct

effect is present, which is negative. Here, an absolute decrease of 0.20% in default

probabilities causes a relative change in the total loan volume by roughly −0.4%.

As outlined by (4.3.3), the bank owners choose, given any of the confidence levels

0.2%, 0.5%, or 1.0% and given the value of the success probabilities, those structural

confidence levels that result in v(p, `V ) = mp2α2(1− `V )− (1−mp1)α1`V . Thus the

total feasible deposit volume strictly increases in ` for ` 6= α2

α1+α2, setting incentives

to choose higher loan-allocation rates than the bank owners do without regulation.

Indeed, this is true for p = 1.0%, but only partly for p = 0.5% and never for

p = 0.2%: from m = 0.9984 on, the unregulated bank owners offer a loan-allocation

rate higher than α1

α1+α2.

Mostly, the household benefits from these higher loan-allocation rates in terms of

deposit return volatilities if p = 1%. The return volatility of the bank’s loan portfolio

reduces for p = 1%, too, as Figure 4.18 shows.


0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062

890.1870.1850.1830.1810.1899.0879.0859.0839.0819.0seitivitcudorp'smrifhtobrofreilpitlumnommoC

Tota

l loa

n vo

lum

enoitalugero/w

%1 level



1=2c,5.0=1c

ehttaRaV

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.19: Total loan volume as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium total loan volumes L on shifts in eachproject’s productivity αi. Shifts in productivity are modeled by a common multiple m. Total loanvolumes are shown for four different regimes: the laissez-faire equilibrium, and the StandardizedApproach with both equal and different risk weights, and the VaR approach with confidence levelsp = 1% and p = 0.5%. Cf. Buhler/Koziol/Sygusch (2008, Fig. 5, p. 156).

4.4 Productivity Shocks

4.4.1 Total Volumes and Cyclical Dampening

Finally, let us consider changes in productivity modeled by jointly scaling the

parameters αi. Higher productivity is associated with more favorable economic

conditions: total output increases given the inputs and given risks: the economy as

a whole is better-off.

The productivity multiplier m is bounded from below by the condition Rf < p1α1m,

i.e. by Condition (3.2.6), and from above by α2m ≤ 2, i.e. by Condition (3.2.7),

hence altogether byRf

p1α1

< m ≤ 2

α2

.

The lower bound for the multiplier is identical to the lower bound of the probability

multiplier as both multipliers affect the projects’ expected gross returns identically.

The upper bound is due to (3.2.7), an assumption that is not necessary for the model

to work, but that has been helpful to derive some general results. In this example,

4.4. PRODUCTIVITY SHOCKS 147

the household is already constrained by its initial wealth WH from m = 1.156 on

whereas the upper bound allows for values of the multiplier up to 1.666. If WH

had been set equal to 4000, the household would have been constrained from 1.266

on. In between, that is for 1.156 ≤ m ≤ 1.266, the unregulated economy would

qualitatively behave as it does for 0.958 ≤ m ≤ 1.156. In particular, both the

Standardized Approach and the VaR-approach at all three levels of confidence are

binding.

Though the multiplier concerning the gross returns αi shifts the projects’ expected

returns as the multiplier concerning the probabilities pi and q does, a fundamental

difference is that the latter multiplier has been assumed to affect the projects’

correlation. Another difference shows up in the sensitivity of the total loan/deposit

volume in equilibrium: it exhibits a higher sensitivity to changes in the probabilities

pi and q than to changes in the gross returns αi. Furthermore, the equilibrium

total loan volumes in the unregulated economy and under the VaR-requirements at

0.5% and at 1%, are concave in the multiplier for both gross returns, but convex in

the multiplier for the single and the joint success probabilities within each domain

between the jumps.

Low values of the multiplier m mean low revenues for the firms in case their projects

succeed and thus lower expected loan redemptions. In the end, lower revenues to

both firms imply lower residual values to the household if the bank defaults. In

short, lower values of this multiplier m reduce the household’s willingness to save

its funds by means of bank deposits. It will rather raise the weight of the risk-

free asset in its portfolio. If m rises, the opposite is true. Figure 4.19 and 4.20,

respectively, show that the unregulated total loan/deposit volume strictly increases

with increasing productivity.

As both forms of regulations result in fixing a maximum lending amount and as the

unregulated bank strictly increases its total loan/deposit volume in the productivity

multiplier m, regulation becomes binding from a given threshold value of m on.

Consequently, regulation does not have a pro-cyclical impact on total lending.

Comparing pairwise Figure 4.19 to Figure 4.12 and Figure 4.20 to Figure 4.11

shows that the various regulatory regimes show similar patterns concerning cyclical

effects in both scenarios, in the one with joint changes in the probabilities pi and q

and in the one with joint changes in the gross returns αi. Furthermore, there are

some similarities concerning the pairwise overlapping behavior of total loan volumes

resulting from different required confidence levels. This behavior will be discussed

in Subsection 4.4.3.


0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062


Tota

l loa

n vo

lum

enoitalugero/w



1=2c,5.0=1c

ehttaRaVlevel%2.0

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.20: Total loan volume as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium total loan volumes L on shifts in eachproject’s productivity αi. Shifts in productivity are modeled by a common multiple m. Total loanvolumes are shown for four different regimes: the laissez-faire equilibrium, and the StandardizedApproach with both equal and different risk weights, and the VaR approach with confidence levelsp = 0.5% and p = 0.2%.

4.4.2 Effects under the Standardized Approach

If both loans are equally weighted under the Standardized Approach there is no

reaction of the total loan volume to changes in the productivity as long as the

regulation is binding. It is fixed to 1c·c1 ·WB. Thus, the regulated total loan volume

is not pro-cyclical. Furthermore, both loans and thus state-contingent deposit

redemptions become so attractive with increasing productivity that the bank shifts

its funds in favor for the riskier Loan 2. Despite the increasing risk associated with

the loan portfolio under binding regulation, the deposit interest rate RSD is lower than

the interest promised by the unregulated bank, R∗D. Moreover, RSD strictly decreases

in the multiplier m, while R∗D strictly increases, thus sustaining the growing deposit

volume in the case without regulation. As a result, the deposit interest rate is

counter-cyclically affected by regulation.

Figure 4.21 (4.22, respectively) shows that the volume granted as a loan to Firm

1, LS1 , strictly decreases in its expected gross return, while Figure 4.23 (4.22,

respectively) shows that the volume granted to Firm 2 strictly increases in the

productivity multiplier. From m = 1.195 on, the volume granted to Firm 2 exceeds


0

002

004

006

008

0001

0021

0041

0061

0081

52.132.112.191.171.151.131.111.190.170.150.11mriFotnaolnonruterssorgdetcepxE

Loan

1

%1 level

noitalugero/w



1=2c,5.0=1c

ehttaRaV

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.21: Loan to Firm 1 as a function of its expected gross return

This figure illustrates the dependence of the equilibrium loan volumes granted to Firm 1 on itsexpected gross return. Changes in its expected gross return p1α1m are caused by m where m jointlyscales each project’s productivity αi. The loan volumes L1 are shown for four different regimes: thelaissez-faire equilibrium, and the Standardized Approach with both equal and different risk weights,and the VaR approach with confidence levels p = 1% and p = 0.5%.

that granted to Firm 1. From m = 1.458 on, the bank’s choice corresponds to Case

2 while it has been compatible to Case 1 below. Both these thresholds are not shown

in the figures considered.

Although the loan granted to the high-risk firms grows under a binding regulation,

its volume is not pro-cyclically affected, whereas the loan to the low-risk firm shows

a counter-cyclical pattern. Thus the regulated bank will build up more risk in a

boom period compared to an unregulated bank, which also translates into a higher

return volatility of loan redemptions, as shown by Figure 4.26. Regulation by equal

risk weights only dampens the return volatility of the deposit pay-off.

This example illustrates that risk-insensitive regulation may induce banks to relax

their lending standards in good times, resulting in an excessive build-up of risks

that may materialize some periods later and thus worsen a downturn. Therefore,

regulation may affect lending and the size of the real economy in a pro-cyclical

manner.

This type of counter-cyclical risk-taking has also been empirically documented.

Ayuso/Perez/Saurına (2004) have analyzed this phenomenon with Spanish banks.


0

002

004

006

008

0001

0021

0041

0061

0081

52.132.112.191.171.151.131.111.190.170.150.11mriFotnaolnonruterssorgdetcepxE

Loan

1noitalugero/w



1=2c,5.0=1c

ehttaRaVlevel%2.0

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.22: Loan to Firm 1 as a function of its expected gross return

This figure illustrates the dependence of the equilibrium loan volumes granted to Firm 1 on itsexpected gross return. Changes in its expected gross return p1α1m are caused by m where m jointlyscales each project’s productivity αi. The loan volumes L1 are shown for four different regimes: thelaissez-faire equilibrium, and the Standardized Approach with both equal and different risk weights,and the VaR approach with confidence levels p = 0.5% and p = 0.2%.

They find that the accumulation of risks is reflected by counter-cyclical capital

buffers, a relation that has been reported for many other countries and jurisdictions

as well.7 As a matter of fact, these studies were still carried out under the dated Basel

I Accord where lower default probabilities could not affect the volume of regulatory

capital needed.8 Crude risk weighting, as it is proposed by the Standardized

Approach, however, will show a comparative sluggishness toward changes in credit

risk.

The phenomenon of counter-cyclical risk-taking could also be aligned with the notion

that loan officers could be oblivious. This hypothesis, the so-called “institutional

memory hypothesis”, was confirmed by Berger/Udell (2004) for United-States based

banks during the eighties and nineties.

If a risk weight of c1 = 0.5 is assigned to Loan 1 whereas Loan 2 is still weighted by

c2 = 1, the bank can increase its total loan volume by increasing the volume granted

7Cf. Bikker/Metzemakers (2007) for OECD countries, Jokipii/Milne (2006) for pan-Europeansamples, Lindquist (2004) for Norway, and Stolz/Wedow (2005) for German savings andcooperative banks.

8By the new Accord, however, through-the-cycle or long-run credit ratings are requested bysupervisory authorities, cf. BCBS (2004, Art. 447).


002

003

004

005

006

007

008

009

0001


Loan

2

%1 level

noitalugero/w



1=2c,5.0=1c

ehttaRaV

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.23: Loan to Firm 2 as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium loan volumes granted to Firm 2 on shifts ineach project’s productivity αi. Shifts in productivity are modeled by a common multiple m. The loanvolumes L2 are shown for four different regimes: the laissez-faire equilibrium, and the StandardizedApproach with both equal and different risk weights, and the VaR approach with confidence levelsp = 1% and p = 0.5%.

as a loan to Firm 1. As a consequence, the total loan volume strictly increases

in the firms’ productivity and we observe the opposite to the case of equal risk

weights concerning single loan volumes under a binding regulation. That is, the

low-risk firm benefits from increasing productivity, but not as much as it does if the

bank is unregulated. The volume granted as a loan to the riskier firm, Firm 2, is

counter-cyclical under binding regulation. Figure 4.26 shows that regulation lowers

the return volatility on the bank’s loan portfolio if c1 = 0.5 and c2 = 1 holds, in

contrast to the case of equal risk weights.

4.4.3 Effects under the Value-at-Risk Approach

Under the VaR approach the picture is much more diverse than under the

Standardized Approach. Apart from some jumps, the total loan volume is not

pro-cyclically affected by regulation if the confidence levels p are set equal to 0.5%

or 1.0%. For p = 0.2%, the total loan volume is counter-cyclically affected by

binding regulation (cf. Fig. 4.20). Single-loan volumes can be both pro- or counter-

cyclical, depending on the level of confidence but also on the level of productivity.


002

003

004

005

006

007

008

009

0001


Loan

2noitalugero/w



1=2c,5.0=1c

ehttaRaVlevel%2.0

ehttaRaVlevel%5.0

1=2c=1c

Figure 4.24: Loan to Firm 2 as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium loan volumes granted to Firm 2 on shifts ineach project’s productivity αi. Shifts in productivity are modeled by a common multiple m. The loanvolumes L2 are shown for four different regimes: the laissez-faire equilibrium, and the StandardizedApproach with both equal and different risk weights, and the VaR approach with confidence levelsp = 0.5% and p = 0.2%.

Furthermore, single loan volumes can outweigh their unregulated counterparts in

volume, as illustrated in Figures 4.21 to 4.23.

In what follows, the effects under different confidence levels are discussed separately,

starting with p = 1% and proceeding to the other required levels of confidence in

descending order.

If p = 1% is required, regulation becomes binding from m = 0.957 on. Up to

m = 0.985, it is optimal simply to retain 1 − p2 as level of confidence. From

m = 0.986 on, however, the bank switches voluntarily to p = 1 − p1 = 0.5%,

resulting in a loan-allocation rate `V that exceeds `∗ such that the volume granted

to Firm 1 under regulation is higher than the volume granted without regulation,

as Figure 4.21 shows. In the figure, this jump is identified with an expected gross

return on the first firm’s project of approximately 1.13. The volume of LV1 exceeds

that of L∗1 from that threshold on, and LV1 reacts in a pro-cyclical manner to further

increases in productivity. The effects that the change in the level of confidence brings

about not only cause jumps in `V and LV1 , but in the total loan volume LV also,

as illustrated in Figure 4.19. Concerning the loan to Firm 2, LV2 , we observe the


opposite: it exceeds its unregulated counterpart in volume for 0.957 < m ≤ 0.985

and is pro-cyclical on this domain. For m ≥ 0.986, it is even counter-cyclical. In

this area, its volume equals that granted under p = 0.5%, as Figure 4.23 shows.

If the confidence level p equals 1%, the bank is free to meet all structural confidence

levels from p = 1− p1 − p2 + q to p = 1− p2, as p1 > p2 = 99% holds. If it complies

with p = 1− p2, but not with p = 1− p1, the regulatory constraint function v(p, `)

becomes, according to (3.3.46)

v(1− p2, `) = p1mα1`− (1− p2)mα2(1− `) ,

in connection with

`V <α2

α1 + α2

.

In this example, the bank, required to reach 1%, effectively reaches the p = (1−p2)-

level for 0.957 ≤ m ≤ 0.985. On this domain, the regulated bank chooses loan-

allocation rates ranging from 44.74% to 37.10%. In particular,

` >(1− p2)α2

(1− p2)α2 + p1α1

≈ 1.038%

is fulfilled by the parameter values given in Table 4.1, such that v(1 − p2, `) > 0

holds in equilibrium.

But for 0.986 ≤ m ≤ 1.666, the bank voluntarily chooses to meet the stricter level

of p = 0.5%. The corresponding regulatory constraint is therefore

v(1− p1, `) = p2mα2(1− `)− (1− p1)mα1` ,

with

`V >α2

α1 + α2

,

as the domain of appropriate loan-allocation rates. This confidence level results in

loan-allocation rates varying from 68.92% to 94.61%. Therefore,

` <p2α2

p2α2 + (1− p1)α1

≈ 99.52%,

holds in this example, implying v(1−p1, `) > 0 in equilibrium. Thus, the regulatory

constraint function always remains positive if the bank is required to attain p = 1%.

The voluntary change in the confidence level shows that an absolute rise in both

projects’ gross returns can be sufficient to make the risk-neutral bank shift its loan


portfolio in favor of the low-risk loan, in this case Loan 1.

Joint increases in productivity always have a negative direct effect on the eligible

total loan/deposit volume as long as v(p, `) > 0 holds, since

∂DV

∂m= −τ ·WB · ∂v(p,`V )

∂m

[v(p, `V ) + ε]2(4.4.1)

is negative if∂v(p, `V )

∂m=

v(p, `V )

m> 0

holds. The latter is the case here, as shown above. Thus, the indirect effect (4.3.3)

must be positive again and outweigh the direct effect as the total loan/deposit

volume strictly increases in the joint productivity multiplier. Hence,

∂DV

∂`· d`

V

dm> 0

must hold true, whereas we note that both single derivatives are negative for 0.957 ≤m ≤ 0.985, and positive for 0.986 ≤ m ≤ 1.666.

Concerning the volume of total lending under the levels of confidence equal to 1%

and equal to 0.5%, there is a remarkable outcome for 0.957 ≤ m ≤ 0.985. On

this domain, the total loan volume LV under the tighter confidence level p = 0.5%

exceeds the total loan volume under p = 1%. In fact, the bank can attain the less

strict level with a lower loan-allocation rate `, i.e. with a riskier loan portfolio. loan-

allocation rates are 9.87 to 31.49 percentage points lower than where a confidence

level equal to p = 0.5% is required. The higher risk is rewarded to the household by

deposit rates that are raised by 65 to 91 base points. This compensation does not

suffice to assure that the household supplies at least as much deposits as it supplies

under the confidence level p = 0.5%. Despite the lower total loan/deposit volume

and the higher deposit interest rates, the monopolistic bank can still increase its

expected final wealth by 0.09 to 1.35 dollars if it attains p = 1%. For this reason,

the bank does not voluntarily stick to the tighter level of 0.5% to raise a higher

deposit volume. Likewise, if regulation requires 0.5%, the bank can never achieve

the gains it could obtain under the less strict level of 1%.

The VaR approach with confidence level p = 0.5% becomes binding at m = 0.927.

Up to m = 0.952, it is even optimal for the bank to comply with p = 1−p1−p2 +q =

0.2%. If the bank effectively attained the p = (1− p1) level, it would have to choose

loan-allocation rates above α2

α1+α2, cf. (3.3.46). In contrast, compliance with the

stricter level of p = 1 − p1 − p2 + q = 0.2% allows for loan-allocation rates below


40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0


Ret

urn

vola

tiliti

es

noitalugero/w


1=2c=1c)stisoped(

ehttaRaVlevel%2.0

)stisoped(



1=2c=1c)snaol(

ehttaRaVlevel%2.0

)snaol(

Figure 4.25: Return volatilities as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium return volatilities on shifts in each project’sproductivity αi. Shifts in productivity are modeled using a common multiple m. Solid and dashedlines represent volatilities of returns on deposits, σD. Diamonds and crosses represent volatilities ontotal loans, σ. Return volatilities are shown for three different regimes: the laissez-faire equilibrium,the Standardized Approach with both risk weights equaling one, ci = 1, and the VaR approach withconfidence level p = 0.2%.

α2

α1+α2, cf. (3.3.46), thus increasing marginal returns on the loan portfolio. The

unregulated bank attains a level of confidence equal to 1% on this domain and

chooses loan-allocation rates between zero and 41.76%, i.e. also lower than α2

α1+α2.

From m = 0.953 on, the bank chooses `V and RVD such that p = 0.5% is reached.

From m = 0.941 on, the total loan volume is not pro-cyclical, but the volume granted

to Firm 1 is pro-cyclical. Loan 2 behaves counter-cyclically.

For multipliers of 0.927 ≤ m ≤ 0.952, the requirement p = 0.2% leads to the

same results as described for p = 0.5%. From m = 0.953 on, the bank will remain

with `V = α2

α1+α2as this loan-allocation rate allows for the highest feasible total

loan/deposit volume under p = 1−p1−p2 +q = 0.2% (cf. Result 24). Consequently,

only the direct effect is present, and is negative since v(1 − p1 − p2 + q, α2

α1+α2) > 0

holds, cf. (4.4.1). As a consequence, the total loan volume and the single loan

volumes strictly decrease in the multiplier for αi in equilibrium and thus become

counter-cyclical under p = 0.2%.

The effects of the VaR regulation on return volatilities are mixed: with all confidence


40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0


Ret

urn

vola

tiliti

esnoitalugero/w


1=2c=1c)stisoped(


dna5.0=1c)stisoped(1=2c


desidradnatshtiwhcaorppa

)snaol(1=2c=1c


dna5.0=1c)snaol(1=2c

Figure 4.26: Return volatilities as a function of shifts in productivity

This figure illustrates the dependence of the equilibrium return volatilities on shifts in each project’sproductivity αi. Shifts in productivity are modeled using a common multiple m. Solid and dashedlines and dots represent the volatilities of returns on deposits. Diamonds and crosses represent thevolatilities of returns on total loans. Return volatilities are shown for three different regimes: thelaissez-faire equilibrium, and the Standardized Approach with both equal and different risk weights.

levels, the return volatility on the regulated loan portfolio is dampened for rather

low values of the productivity multiplier m, but is enhanced for high m. If p = 1%

is required, the return volatilities of the regulated loan portfolio also exceed its

unregulated counterparts for 0.957 ≤ m ≤ 0.985, i.e. on the domain, where the

bank still only attains a confidence level of p = 1%, and does not yet reach the

tighter level of p = 0.5%.

If the bank must satisfy p = 1% or p = 0.5%, the deposit return volatility is

dampened given relatively low values for the firms’ productivity, but it becomes

enhanced for rather high multipliers as well. It is only under a required confidence

level of p = 0.2% that the deposit return volatility is reduced for all levels of

productivity. The effects of the VaR-based regulation at p = 0.2% on return

volatilities are shown in Figure 4.25.

Comparing loan volumes granted under the VaR approaches to those granted under

the Standardized Approaches does not yield a clear-cut picture concerning pro-

cyclicality either.

Results can be best summarized by stating that volumes and volatilities react more

4.5. SUMMARY 157

strongly to shocks under regulation the more leeway regulation concedes to banks.

Under the most rigid forms of regulation, such as equal risk weights or the tightest

eligible confidence level, volumes are the most impervious to shocks. Rigidity,

however, does not mean that risks are effectively reduced.

4.5 Summary

The theoretical and numerical analyses suggest that the potential of pro-cyclical

effects caused by capital requirements on lending depends mainly on three factors:

the kind and tightness of regulation, the type of shock considered, and the loan

volume considered (i.e. total vs. single volumes).

Total loan volumes are affected pro-cyclically when there are shocks in the bank’s

initial equity because capital requirements, both the Standardized Approach as well

as the VaR approach, are directly linked to the bank’s level of equity. Although the

proportionality factor may change in the loan-allocation rate, it is more or less fixed

to a given value, or at least to a given range of values.

There is no pro-cyclical effect, however, if we consider joint moves in the firm

projects’ success probabilities or in the firm projects’ productivity. Under the

tightest level of confidence enabling positive total loan volumes, the regulated total

loan volume is counter-cyclical concerning rising productivity. In particular, a joint

rise in success probabilities or in productivity always have a direct negative impact

on the feasible total loan/deposit volume under the VaR approach.9

Concerning the single loan volumes, results are much more diverse. Under all forms

of capital adequacy rules considered, single loan volumes may react pro-cyclically,

non-pro-cyclically, or even counter-cyclically to any kind of shocks. Under binding

VaR-requirements either single loan volume may also outweigh its unregulated

counterpart. With the parameter values considered, this phenomenon cannot be

confirmed under the Standardized Approach as it reduces the feasible total loan

volume too strongly. Similarly, Repullo/Suarez (2004) show within a one-period

bank model that Basel II is more favorable for borrowers than Basel I in terms of

loan interest rates.

Besides the issues of cyclicality, this study gives some insights into how the bank’s

risk-taking can be affected by regulation.

9Note that a joint rise in productivity leads to a reduction in the feasible total loan/depositvolume under the VaR approach if and only if v(p, `) > 0 holds.


First, let us consider the loan-allocation rate. Equal risk weights under the

Standardized Approach result in lower loan-allocations rates than if the bank is

unregulated. Thus, the high-risk loan becomes relatively more important to the

bank: lower business volumes are compensated by high-yield, and highly risky

investments.

If the risk weights differentiate according to the default risk of the loans, the bank

will choose higher loan-allocation rates under regulation than without regulation.

Under the VaR approach, the relation between `V and `∗ depends on the confidence

level held which can be different from, i.e. tighter than, the confidence level required.

With p = 1 − p2, the regulated bank always chooses lower, with p = 1 − p1 always

higher loan-allocation rates than without regulation. Things are mixed under p =

1 − p1 − p2 + q as the bank mostly chooses the intermediate level of α2

α1+α2, hence

the equilibrium outlined by Result 24.

The regulated bank devotes a higher fraction to the more risky, i.e. more profitable

loan, for the following two distinct reasons: under the Standardized Approach with

fixed risk weights, the bank has to reduce its loan volumes and hence must forgo

profit opportunities under a binding regulatory constraint. Contrary to risk weights

accounting for differences in the default probability, there is no possibility to weaken

the regulatory constraint on total volumes. The forgone profit opportunities by this

volume reduction are therefore offset by increasing the marginal return, and hence

the credit risk.

Under the VaR approach, this adverse selection occurs if the bank must comply

with p = 1 − p2, always preferring Loan 2. Thus, firms featuring more risk in

their production plans might even face situations where they have a comparative

advantage among their less risky competitors on the market for bank loans.

Volatilities of returns on the loan portfolio can become both higher or lower through

regulation, irrespective of the form of capital rule considered. The same applies to

the volatilities of returns on deposits. Thus, risks are rather contained by volume,

not by structure.

Yet, particularly those results concerning risk-taking under the VaR approach may

be driven by the Bernoulli distribution. If the redemptions of the whole loan portfolio

are modeled by a continuous distribution, the jumps in loan volumes and loan-

allocation rates should vanish for two reasons: first, Case migrations can no longer

occur and second, changes in confidence levels under a VaR approach are no longer

achieved by jumps in the loan-allocation rate. In the following part we therefore

4.5. SUMMARY 159

present a variation of the model that allows for a normal approximation of aggregate

loan redemptions.


Part III

Normally Distributed Loan

Redemptions

161

Chapter 5

A Basis for the Normal

Distribution

5.1 Introduction

We extend the analysis of Chapter 3 and 4 to more than two firms. More precisely,

there are two groups of firms, each containing n > 1 firms. As in the Bernoulli

framework, each firm undertakes one single risky project that can either be successful

or fail. Each project’s return depends on the returns of some other projects.

We assume that the number of projects which any single project outcome can

be correlated with may be arbitrarily high, but fixed. Thus, the depth of the

dependence structure does not expand with the number of projects and sector-

wise returns can be approximated by the normal distribution. The set-up on the

firm level builds up on a random probability model as it can be found in its general

form in Joe (2001). The normal approximation simplifies the analysis compared to

that performed in the preceding part, Part II, as now one can dispense with the

distinction of different deposit redemption cases.

The following sections will present the underlying dependence structure that allows

for an approximation by the normal distribution. Chapter 6 discusses the bank’s

and the household’s objectives in an analogous setting, as can be found in Chapter

3 and as sketched in Chapter 2. Section 6.6 analyzes the impact through regulation

on the bank’s lending behavior in a similar way as performed in Chapter 4. Chapter

5 and 6 are based upon Buhler/Koziol/Sygusch (2007), but considerably extend

their analysis, in particular concerning the numerical study of regulatory impacts

on lending and the exposition of the underlying dependence structure.

163

164 CHAPTER 5. A BASIS FOR THE NORMAL DISTRIBUTION

5.2 Firms and Sectors

There are, analogous to Section 3.2.1, two types of projects i whose outcomes are

Bernoulli-distributed, and which can be differentiated according to their probability

of success and two types of firms that can be differentiated according to their

technology. Assumptions (3.2.5), (3.2.6), and (3.2.7) are supposed to hold as

well, unless we explicitly refer to the case of equality. A given technology and a

given project can either be perfect complements that fit together for production, or

completely useless. As each firm is assumed to have a fixed technology, they have

no choice about the project they undertake. Consequently, there is ex ante a group

of firms running project i = 1 (with technology i = 1) and another group of firms

running project i = 2 (with technology i = 2). Each of both groups is referred

to as a sector. Both sectors are assumed to be equal in size such that each sector

contains n firms. Firms are indexed by j. X(j)i denotes the random success of a

single project i undertaken by firm j, j = 1, . . . , n. The outcome of each project is

Bernoulli-distributed, i.e. a project is either successful, X(j)i = 1, or it fails, X

(j)i = 0.

In addition to Chapter 3, the success of undertaking a single project does not only

depend on the performance of one other firm, but potentially on the performance

of many other firms. Thus, each project, and in turn the redemption of each single

loan, depends on a number of risk factors. These risk factors affect the actual

outcome of each project by influencing the outcome of the corresponding success

probability p(j)i that characterizes the risk-return profile of project i undertaken

by firm j. Therefore, firms can be differentiated by their idiosyncratic risk within

each sector i. Thus success probabilities are random variables and each Bernoulli-

distribution can formally be stated as

X(j)i ∼ B(p

(j)i ), P(X

(j)i = 1) = p

(j)i , i = 1, 2, j = 1, . . . , n . (5.2.1)

This set-up is known as Bernoulli-mixture model (Joe, 2001).

More specifically, consider firm j, j 6= 1, n, from sector i. Let its success probability

p(j)i be exposed to firm j’s idiosyncratic factor ξ

(j)i and to the idiosyncratic risk of its

follower within the same sector, i.e. to ξ(j+1)i . This common risk factor represents

common demand shocks to which the respective firms may be exposed to as they

produce similar products or operate in the same geographical region. Likewise, the

firm shares risks with its predecessor, reflected by ξ(j−1)i . Hence, firm j from sector

i shares the risk loading ξ(j)i with both its predecessor and its follower. Moreover,

the firm considered also faces some risks stemming from the other sector which can

5.2. FIRMS AND SECTORS 165

be interpreted by supply chain relations. Specifically, we assume that it is exposed

to the idiosyncratic risk of firm j from the other sector, labeled 3 − i, i.e. to the

risk factor ξ(j)3−i. The assumption that projects cannot be correlated with any other

project too far away points out that even firms producing similar products do not

necessarily suffer from the same shock the same time. To represent the preceding

notions, we set up the following model for each probability p(j)i ,

p(j)i ≡

1− b(1)i · ξ(1)

i − c(2)i · ξ(2)

i − d(1)i · ξ(1)

3−i , j = 1

1− a(j−1)i · ξ(j−1)

i − b(j)i · ξ(j)

i − c(j+1)i · ξ(j+1)

i − d(j)i · ξ(j)

3−i , j = 2, . . . , n− 1

1− a(n−1)i · ξ(n−1)

i − b(n)i · ξ(n)

i − d(n)i · ξ(n)

3−i , j = n

(5.2.2)

for i = 1, 2 where a(j)i , b

(j)i , c

(j)i , d

(j)i are fixed, strictly positive numbers and where

ξ(j)i are mutually independent, uniformly distributed random variables with ξ

(j)i ∼

U(0, 2t) for all i = 1, 2, j = 1, . . . , n. Hence, the success probabilities only take

values between zero and one if furthermore

2 · t · (a(j)i + b

(j)i + c

(j)i + d

(j)i ) < 1 (5.2.3)

is fulfilled. Their support is thus given by1

supp(p(j)i ) =

[1− 2 · (b(1)

i + c(2)i + d

(1)i ) · t, 1

], j = 1

[1− 2 · (a(j−1)

i + b(j)i + c

(j+1)i + d

(j)i ) · t, 1

], j = 2, . . . , n− 1

[1− 2 · (a(n−1)

i + b(n)i + d

(n)i ) · t, 1

], j = n

,

hence, supp(p(j)i ) ( (0, 1].

For simplicity, we assume that the parameters a(j)i , b

(j)i , c

(j)i , d

(j)i are only sector-

specific, not firm-specific. The expected probability of success, E[p(j)i ] = pi ,

simplifies to

p(j)i =

1− (bi + ci + di) · t , j = 1

1− (ai + bi + ci + di) · t , j = 2, . . . , n− 1 ,

1− (ai + bi + di) · t , j = n

i = 1, 2 .

As a consequence, the success of firm j from sector i has as expectation value and

1Under the base case outlined in Table 5.1, the support is [0.99, 1] for p(j)1 and [0.98, 1] for p(j)

2 ,respectively, with j = 2, . . . , n− 1.


variance

E[X(j)i ] = p

(j)i , and V[X

(j)i ] = p

(j)i · (1− p(j)

i ) , i = 1, 2, j = 1, . . . , n , (5.2.4)

respectively, analogous to any generic Bernoulli random variable. Details are given

by (B.1.3) to (B.1.5) in Appendix B.1. By

pi ≡ 1− (ai + bi + ci + di) · t (5.2.5)

we denote to the expected success probabilities for firms j = 2, . . . , n − 1. This

probability is linked to the generic firm j from sector i whose success will characterize

the limit properties of aggregate loan redemptions.

The factor model (5.2.2) implies that within each sector a given firm’s outcome is

correlated with the outcomes of its next two followers and its last two predecessors,

that is

Corr(X(j)i , X

(j±1)i ) =

1

3· (ai + ci)bit

2

pi(1− pi), 1 ≤ j ± 1 ≤ n , (5.2.6)

Corr(X(j)i , X

(j±2)i ) =

1

3· aicit

2

pi(1− pi), 1 ≤ j ± 2 ≤ n . (5.2.7)

Each firm’s undertaken project is also correlated with the projects’ outcomes of the

three neighboring firms from the other sector, hence

Corr(X(j)i , X

(j−1)(3−i) ) =

1

3· (aid3−i + c3−idi)t

2

√p1(1− p1)p2(1− p2)

, 2 ≤ j ≤ n , (5.2.8)

Corr(X(j)i , X

(j)(3−i)) =

1

3· (bid3−i + b3−idi)t

2

√p1(1− p1)p2(1− p2)

, 1 ≤ j ≤ n , (5.2.9)

Corr(X(j)i , X

(j+1)(3−i) ) =

1

3· (cid3−i + a3−idi)t

2

√p1(1− p1)p2(1− p2)

, 1 ≤ j ≤ n− 1 .(5.2.10)

The derivation of these formulæ is shown in Appendix B.1 which follows the

expositions by Joe (2001, pp. 211, 219-220). Note that model parameters cannot

be chosen arbitrarily, but mixture models have the advantage that consistent, non-

trivial correlation structures can be set up for a large number of random variables in a

convenient way.2 Furthermore, mixture models are by far less parameter-consuming

than it is the case with contingency tables.3

2Chaganty/Joe (2006) characterize restrictions on correlations matrices for binary randomvariables and point out that there are positive-definite correlation-matrices that are not compatiblewith any set of univariate success probabilities.

3The number of parameters increases only linearly in the number n of random variables

5.2. FIRMS AND SECTORS 167

The intra- and inter-industrial linkages arising from this model are illustrated by

Figure 5.1. Each horizontal and each vertical arrow represents a common risk factor

with the respective neighbor. These dependencies result in the correlations given by

(5.2.6) and (5.2.9). Diagonal arrows and bowed arrows represent correlations that

are based on common risk weights via common neighbors, cf. (5.2.7), (5.2.8), and

(5.2.10).

Given the preceding model, let us re-write each firm’s objective in a complete analogy

to Section 3.2.1. Particularly, let L(j)i be the loan volume granted to firm j from

sector i and R(j)i be the gr oss interest rate on this loan. Technology is linear and

sector specific such that each firm j from sector i can multiply each dollar invested

by αi > 1 in case of success. The risky redemption from the loan contract is thus

given by

L(j)i = min

L

(j)i ·R(j)

i , αi · X(j)i · L(j)

i

,

and expected final wealth by

E[max

αi · X(j)

i · L(j)i − L(j)

i ·R(j)i , 0

]=

E[p

(j)i ·max

αi −R(j)

i , 0· L(j)

i + (1− p(j)i ) · 0

]= p

(j)i ·max

αi −R(j)

i , 0· L(j)

i ,

being equal to (3.2.2) except for the expectations formed about the success

probability. Because firms are risk-neutral and because firms are not endowed with

any equity initially, they will demand debt finance as given by (3.2.3). Consequently,

the bank optimally sets the loan interest rates according to R∗i = αi for any loan

granted to a firm from sector i and the average loan volume in sector i amounts to

L∗i =1

n· Li, i = 1, 2, j = 1, . . . , n, (5.2.11)

given that the bank lends an aggregate loan volume of Li to the whole sector i.

Again, the aggregate loan volume will have to be determined by the refinancing

opportunities that the household offers to the bank.

considered if mixture models are used whereas the approach by contingency tables requires 2n− 1parameters (cf. Joe, 2001, p. 211).


X(1)2 X

(2)2 X

(3)2 X

(4)2 X

(5)2 X

(6)2 · · · · · · X

(n)2

X(1)1 X

(2)1 X

(3)1 X

(4)1 X

(5)1 X

(6)1 · · · · · · X

(n)1

*Y *Y *Y *Y *Y *Y· · ·

j j j j j j· · ·

· ·· ··

@@@R@@@I @

@@R@@@I @

@@R@@@I @

@@R@@@I @

@@R@@@I @

@@R@@@I

· ·· ·

@@@R@@@I6

?6?

6?

6?

6?

6? ··

···

·····

6?

- - - - - - · · · · · -

- - - - - - · · · · · -

Figure 5.1: Correlation structure between projects

This figure illustrates the dependence structure between project successes if the success probabilitiesare given by (5.2.2). Each arrow represents a positive correlation between the respective projects.

5.3 The m-Dependence Structure and Its Limit

Distribution

The aim of this section is to characterize this dependence structure further and to

present a framework that enables the derivation of the limit distributions for the

sums of sector-wise loan repayments∑n

j=1 L(j)i and for the common repayments.

Beyond the classical central limit theorem that particularly presumes mutual

independence of the random variables Xj considered within the sequence Xji=j,...,n,

Serfling (1968, pp. 1158f) presents the following conditions,

limn→∞

E

(

1√n·a+n∑

j=a+1

(Xj − E

[Xj

]))2 = s2, s2 > 0,

uniformly for all a ≥ 0, (5.3.1)

E∣∣∣Xj

∣∣∣2+δ

≤ M, M <∞,for some δ > 0 , (5.3.2)

that are necessary so that the following normalized sum of the random variables Xj,

j = 1, . . . , n,1√n · s ·

n∑

j=1

(Xj − E

[Xj

]),

is asymptotically normal with mean zero and variance equal to one, i.e.

limn→∞

P

[∑nj=1 Xj −

∑nj=1 E[Xj]√

n · s ≤ z

]= Φ(z) ,

where Φ(z) is the cumulative distribution function of the standard normal

5.3. THE M -DEPENDENCE STRUCTURE AND ITS LIMIT DISTRIBUTION 169

distribution. If the random variables are independent, Assumptions (5.3.1) and

(5.3.2) become sufficient and the classical central limit theorem holds.

For dependent random variables, Assumption (5.3.1) may fail in general. (5.3.1)

requires the variance of the sum of Xj, j = 1, . . . , n, not to increase by an order

higher than n. However, V(∑n

j=1 Xj) grows by n2 for general covariance patterns. So

it is necessary to establish a covariance structure among the random variables that

excludes any dependencies between two sets of random variables as soon as these

two sets get too far apart. A notion such as this can be rendered more precisely

by the so-called m-dependence (Hoeffding/Robbins, 1948, p. 773; Billingsley, 1995,

p. 364; and Serfling, 1968, p. 1162):

Definition 1. A sequence Xi = X1, X2, . . . , Xn is said to be m-dependent if

the two sequences Xa−r, Xa−r+1, . . . , Xa and Xb, Xb+1, . . . , Xb+s are independent

sets of random variables if b− a > m for all r, s ≥ 0 and m a fixed constant.

Clearly, 0-dependence is equivalent to independence whereas the higher is the value

for m, the deeper becomes the dependence structure. Note, however, that m is a

fixed constant such that the depth of the dependence structure does not increase in

n. Consequently, the number of covariance terms in the variance of the sum over Xi,

i = 1, . . . , n, increases only linearly in n such that Assumption (5.3.1) is fulfilled.

Now, consider sector i as defined by the factor model (5.2.2) in Section 5.2 and

take X(3)i as a starting point. Then X

(3)i depends on X

(j)i up to j = 5 and

down to j = 1 (cf. Fig. 5.1). Thus the two sequences X(1)i , X

(2)i , X

(3)i and

X(6)i , X

(7)i , . . . , X

(6+s)i , s ≥ 0, and so forth are independent sets of random

variables. By Definition 1, the sequence X(j)i is 2-dependent. Similarly, the bivariate

sequence (X(j)1 , X

(j)2 ) is 2-dependent as well. This 2-dependence structure can

be illustrated by replacing the projects’ returns X(j)i in Figure 5.1 by the factor

loadings ξ(j)i according to (5.2.2).

Given m-dependent sequences, central limit theorems can be applied again, in

particular given the Assumptions (5.3.1) and (5.3.2) (Serfling, 1968, p. 1159). His

theorem resorts to the seminal work of Hoeffding/Robbins (1948). Furthermore, we

can take advantage of the assumption that the random variables are identically

distributed, i.e. of (5.2.1) and (5.2.4). As a direct consequence, we can relax

Condition (5.3.1) in the univariate case (Hoeffding/Robbins, 1948, pp. 774, 776).

Moreover, we present the theorem applicable to the bivariate case which can, in turn,

be readily stated for the univariate case (ibid. pp. 776, 776-777 and cf. Billingsley,

1995, p. 364, who extends the univariate case to so-called α-mixing distributions):


Theorem 1 (Hoeffding/Robbins, 1948). Let (X(j)1 , X

(j)2 ) =

(X(1)1 , X

(1)2 ), (X

(2)1 , X

(2)2 ), . . . , (X

(n)1 , X

(n)2 ) be an m-dependent sequence of random

vectors which are identically distributed and whose expectations are normalized to

zero,(

E(X(j)1 ),E(X

(j)2 ))

= (0, 0), and let E|X(j)1 |3 < ∞ and E|X(j)

2 |3 < ∞ hold.

Then as n → ∞, the random vector 1√n

(∑nj=1 X

(j)1 ,∑n

j=1 X(j)2

)has a limiting

normal distribution with mean (0, 0) and covariance matrix

(s2

1 s1,2

s1,2 s22

)

where

s21 = E(X

(1)1 X

(1)1 ) + 2 ·

m∑

j=1

E(X(1)1 X

(j+1)1 ) , (5.3.3)

s1,2 = E(X(1)1 X

(1)2 ) +

m∑

j=1

[E(X

(1)1 X

(j+1)2 ) + E(X

(j+1)1 X

(1)2 )], (5.3.4)

s22 = E(X

(1)2 X

(1)2 ) + 2 ·

m∑

j=1

E(X(1)2 X

(j+1)2 ) . (5.3.5)

Based on this theorem, we will approximate the distribution of aggregate loan

redemptions, both sector-wise and in total, in the following section.

5.4 The Limit Distributions of the Aggregate

Loans

For sufficiently large, but still finite n we may neglect the first and the n-th firm

belonging to each sector. Thus, aggregate loan redemptions Li,

Li ≡n∑

j=1

L(j)i ≡

n∑

j=1

αi · X(j)i · L(j)

i , i = 1, 2 , (5.4.1)

consist of identically distributed loan redemptions on the micro-level. Treating each

loan redemptions L(j)i as being equal to each other within each sector i, i.e. L

(j)i =

L(k)i for all j 6= k, i fixed, facilitates computations of aggregate statistical moments,

and, in addition, makes Theorem 1 applicable. Therefore, we can approximate the

5.4. THE LIMIT DISTRIBUTIONS OF THE AGGREGATE LOANS 171

distribution of the non-centered sums

1√n·

n∑

j=1

X(j)i

by the normal distribution

N (√npiαi, s

2i )

where√npiαi = 1√

n· E[∑n

j=1 X(j)i ], and where si is given according to (5.3.3) and

(5.3.5), respectively. Furthermore, the tuple of both sums is approximately bivariate

normal,

1√n·(

n∑

j=1

X(j)1 ,

n∑

j=1

X(j)2

)>d∼ N (

√n ·(p1α1

p2α2

),

(s2

1 s1,2

s1,2 s22

)) .

To obtain the limit distributions for both loan portfolios, we scale the sums shown

above by√n · Li. Thus, we obtain the aggregate loan redemptions as defined by

(5.4.1) with the following approximate limiting distribution:

Li ≡n∑

j=1

L(j)i ≡

n∑

j=1

X(j)i · L(j)

id∼ N ( piαi · Li,

1

n· s2

i · L2i ) . (5.4.2)

Analogously for the bivariate case,

(L1, L2

)> d∼ N (

(p1α1 · L1

p2α2 · L2

),

1

n·(

s21 · L2

1 s1,2 · L1 · L2

s1,2 · L1 · L2 s22 · L2

2

)) , (5.4.3)

approximately holds. Concerning the last two derivations, we have made use of the

Property (5.2.11) that is valid for the optimal loan volumes granted by the bank.

Thus, each dollar invested into loan portfolio i yields a gross return xi that is

approximately normally distributed with mean µi and variance σ2i ≡ s2i

n,

xi ∼ N (µi, σ2i ) , (5.4.4)

where µi and σi are approximately given by

µi = pi · αi ,

σi ≡si√n

=1√n·√pi · (1− pi) +

2

3(aici + aibi + bici)t2 · αi , i = 1, 2 ,

(5.4.5)


Table 5.1: Parameter values of the base case, Bernoulli mixture model

This table reports the parameter values of the base case for the factor model given by (5.2.2). TheBernoulli distribution characterized by the success probabilities p(j)

i takes values for its expectationand its variance that equal those of the Bernoulli distribution used in Part II under the parametervalues given in Table 4.1.

a1 b1 c1 d1 a2 b2 c2 d2 t

0.02 0.05 0.02 0.01 0.04 0.10 0.04 0.02 0.05

and where pi is given according to (5.2.5). Let the gross return on the sum of both

loan portfolios be given by

x = ` · x1 + (1− `) · x2 , (5.4.6)

where ` is defined in complete analogy to (3.2.14) and, in conjunction with the model

considered in this part, referred to as the portfolio-allocation rate.

Consequently, the return x is normally distributed, x ∼ N (µ, σ2), with

µ = ` · µ1 + (1− `) · µ2 ,

σ2 = `2 · σ21 + 2 · ` · (1− `) · σ1,2 + (1− `)2 · σ2

2 .(5.4.7)

where σ1,2 is the covariance that is given by

σ1,2 ≡s1,2

n=

1

3n· [(a2 + b2 + c2) · d1 + (a1 + b1 + c1) · d2] · t2 · α1α2 .

(5.4.8)

Thus, aggregate loan redemptions∑n

i=j L(j)i and aggregate gross returns xi,

respectively, are approximately correlated by

ρ =13· [(a1 + b1 + c1) · d2 + (a2 + b2 + c2) · d1] · t2

2∏i=1

√pi(1− pi) + 2

3(aibi + aici + bici)t2

. (5.4.9)

Concerning these formulæ we refer to Appendix B.2 where their derivations are

demonstrated and the respective formulæ considering the aggregate loan repayments

Li are given by (B.2.3) to (B.2.6).

For illustrative purposes, we consider our Bernoulli mixture model for a given set of

parameter values, as shown in Table 5.1 and compare the limit distribution with a


Table 5.2: Correlation values of the base case, Bernoulli mixture model

This table reports the values for the project return correlations under the base case. Both inter-and intra-sectoral correlations are shown. Numbers are rounded to three leading digits.

Corr(X(j)1 , X

(j+1)1 ) Corr(X(j)

1 , X(j+2)1 ) Corr(X(j)

2 , X(j+1)2 ) Corr(X(j)

2 , X(j+2)2 )

0.000335 0.0000670 0.000340 0.000680Corr(X(j)

1 , X(j−1)2 ) Corr(X(j)

1 , X(j)2 ) Corr(X(j)

1 , X(j+1)2 )

0.0000479 0.000120 0.0000479

simulated distribution.

The parametrization in Table 5.1 results in p1 = 0.995 and p2 = 0.99 and thus

each single loan redemption has the same distribution as considered in Table 4.1.

However, correlations (cf. Formulæ (5.2.6) to (5.2.10)) are lower, in fact close to zero,

as illustrated in Table 5.2. On the other hand, loan-return correlations are always

well below their associated asset-return correlations unless the latter approaches one

as emphasized by Gersbach/Lipponer (2003, p. 365f).4

Let us assume that each sector consists of n = 40, 000 firms. For simplicity, we

normalize every single loan volume Li to one, such that the aggregate loan volume

Li equals the number n of borrowers. Running the simulation 200 times leads to

the distributions with moments, as shown in Table 5.3. These simulated values are

compared to the theoretical values given by (B.2.3) to (B.2.5) in Appendix B.2 for

parameter values equal to those from Table 5.1.

By this model we are able to approximate the aggregate loan redemptions by the

normal distribution, though intra- and inter-sectoral correlation patterns between

the firms’ projects are present. Inter-sectoral correlations are incurred by the sector-

wise variance as in (5.3.3) and (5.3.5), respectively. Intra-sectoral correlations on

the firms’ level result in the aggregate inter-sectoral correlation as given by (5.3.4).

Particularly, this model can represent a positive correlation between sector-wise

loan redemptions on an aggregate level without losing the central limit property of

normality.

Concerning the equilibrium analysis in this model framework, a smooth limit

distribution has the well-known advantage that we have not to analyze each

loan individually. If we accounted separately for every single of the 2 · n loans,

4Compare also the values from Table 5.2 with the value of the correlation parameter given inTable 4.1, Panel B and the associated discussion of these values considered for the Bernoulli-modelon p. 119.


Table 5.3: Comparison between simulated and limiting distribution

This table reports the values of moments of the random sums∑X

(j)1 , i = 1, 2, i.e. of the sector-

wise loan redemptions. Each sum/sector consists of n = 40, 000 firms as defined by (5.2.1). In thesimulation each sum was drawn N = 200 times. The moments of the limiting normal distributionare given by (B.2.3) to (B.2.5) in Appendix B.2. If necessary, numbers are rounded to six leadingdigits.

statistical moment simulated distribution limiting distribution

E[∑

X(j)1

]≡ µ1 · L1 45, 769.9 45, 770

V[∑

X(j)1

]≡ σ2

1 · L21 253.125 263.389

Jarque-Bera test statistic 1.75630 0

E[∑

X(j)2

]≡ µ2 · L2 47, 522.2 47, 520

V[∑

X(j)2

]≡ σ2

2 · L22 566.898 570.470


Cov[∑

X(j)1 ,∑X

(j)2

]≡ σ1,2 · L1 · L2 0.160122 0.1656

E[

12 ·(∑

X(j)1 +

∑X

(j)2

)]≡ µ · (L1 + L2) 46, 646.0 46, 645

V[

12 ·(∑

X(j)1 +

∑X

(j)2

)]205.086 208.548


22n+1 − 2 · n− 2 Cases would have to be considered as outlined in Section 3.2.2, in

the footnote on p. 51. Finally, the application of the VaR approach is facilitated.

A drawback of this approach is that the gross returns x and, thus, the aggregate loan

portfolio redemptions L, may be negative with probability Φ(−µσ). This probability

decreases in the square root of the number of firms n because of (5.4.5). If the whole

loan portfolio only consists of loans from one sector, L = Li, this probability reduces

to Φ(−µiσi

), i.e.

Φ(−µiσi

)(5.4.5)

= Φ(− piαi ·√n√

pi · (1− pi) + 23(aici + aibi + bici)t2 · αi

)

Table 5.1≈ Φ(− pi√pi · (1− pi)

· √n) .

For large n, this probability becomes negligible. Furthermore, Properties (3.2.5) and

(3.2.6) transfer via (5.4.5) to

µ2 > µ1 > Rf ≥ 1 . (5.4.10)


Fulfilling additionally the inequality a1b1 + a1c1 + b1c1 < a2b2 + a2c2 + b2c2 is thus

a sufficient condition to obtain

σ1 < σ2 . (5.4.11)

Furthermore, following (3.2.7), we set

µi < 2 and by (5.2.3) σi < 0.1 , (5.4.12)

where the latter is obtained if we additionally assume that loan portfolio i consists

of at least 144 borrowers, i.e. n ≥ 144.5 As 0.2 is rather typical for asset-return

volatilities, this bound on σi, which represents the volatility of returns on loan

portfolio i, does not seem to be too restrictive.

As a consequence, we obtain by (5.4.10) and (5.4.12)

µ

σ≥ minµ1, µ2

maxσ1, σ2>

1

0.1= 10 (5.4.13)

for the ratio of expected gross returns over their respective standard deviations,

resulting in the following approximations concerning the density function ϕ(x) and

the cumulative distribution function Φ(x) of the standard normal distribution

ϕ(µσ) ≈ 0, since ϕ(µ

σ) < ϕ(10),

Φ(µσ) ≈ 1, since Φ(µ

σ) > Φ(10),

(5.4.14)

and likewise Φ(−µσ) ≈ 0. Whenever (5.4.14) is used in deriving results, we will

explicitly refer to it. If we do not use this approximation and the positive chance of

negative redemptions of the loan portfolio remains, we assume that these losses are

borne by a governmental authority that is not explicitly modeled.

The one-factor model on which the IRB approach is based can be considered as a

Bernoulli-mixture model with firm default as Bernoulli-distributed random variable

and with the macro factor as the mixing distribution (Giesecke 2004, p. 26). In

contrast to our model, the derivation of the distribution of aggregate defaults (loan

redemptions) rests on a given realization of a single macro factor.6 Thus they

reflect a centralized dependence structure whereas our model pursues a decentralized

dependence structure.

5Condition (5.2.3) allows to bound t by 12 · 1

ai+bi+ci+diresulting in (aici + aibi + bici)t2 <

14 · aici+aibi+bici

(ai+bi+ci+di)2< 1

4 · 12 . Let n ≥ 144, we finally obtain σi ≤ 1

12 ·√

13 · 2 ≈ 0.0962.

6Cf. Giesecke (2004, pp. 21-32), Lutkebohmert (2009), Schonbucher (2000), and BCBS (2004,Art. 272). Gordy (2003) provides a rigorous characterization of these single-factor models.


Chapter 6

The One-Period Model

6.1 Introduction

By the last chapter, Chapter 5, the grounds have been laid for approximating weakly

dependent, Bernoulli-distributed loan redemptions by the normal distribution.

Intra- and inter-sectoral dependencies have been considered on the firm level.

In total, aggregate loan redemptions related to each of both portfolios are

normally distributed and both aggregate loan redemptions preserve the inter-sectoral

dependency.

The model that is considered in this chapter is analogous to that analyzed in Part II.

The two single firms, however, are now replaced by portfolios of which the returns

are normally distributed and which may be correlated, in line with the discussions

in Chapter 5. The risk-averse household supplies deposits and the banks serves as

an intermediary between firms and the household.

The bank’s objective is outlined in the next section, Section 6.2, followed by an

analysis of the household’s decision in Section 6.3. In Section 6.4, the equilibrium

absent of regulation is discussed, in Section 6.5 the equilibrium if the bank is

regulated by a VaR-based approach. In contrast to Part II, the Standardized

Approach is no longer considered as it does not yield any further insights in this

framework considered. The properties of the regulatory constraint as such have been

already discussed in Section 3.3.2.

As a general analysis does not yield further insights into the potential impacts

regulation may have, the analysis is purely based on numerical examples as done

in Section 6.6. The laissez-faire equilibrium is compared to the outcomes under a

VaR-based regulation and partly also to the outcomes arising under the Standardized

177

178 CHAPTER 6. THE ONE-PERIOD MODEL

Approach. Section 6.7 concludes.

6.2 The Bank’s Objective

The bank is risk-neutral and maximizes its expected final wealth. It acts as a

financial intermediary between households and firms. All contracts start at the

beginning of the period considered and mature at the end of the period.

The bank owners can grant loans to firms from two sectors as discussed in Chapter

5. We have already pointed out by (5.2.11) how the bank optimally chooses the loan

interest rates Ri and the loan volumes Li on the individual firm level. The whole

model outlined in the preceding section presumes an equal number of firms in each

sector. However, the bank is still free to assign different total loan volumes Li to

each sector. We assume that this decision on the aggregate level is based on the

bank’s view of the aggregate distribution, in the sense that it considers the limiting

distribution for each sector and for both sectors together. Thus, we can build on the

normal framework, and the return of the bank’s loan portfolio will follow (5.4.6).

The decision variable ` is referred to as the portfolio-allocation rate. Again, there

are no short selling opportunities to the bank, ` ∈ [0, 1]. The bank refinances itself

by fixed, exogenously given equity WB and deposits D supplied by the household.

The deposit contract is a standard debt contract, with the following pay-off function

max x · (D +WB), D ·RD ,

where D is the amount deposited at the bank at the beginning of the period, RD

the contracted gross interest rate on deposits and x the realized gross return on the

loan portfolio. The return on the loan portfolio is given by (5.4.6) and hence x is

normally distributed according to

x ∼ N (µ, σ2) ,

where the expected return µ and the associated volatility σ are given by (5.4.7),

including (5.4.8). In particular, both moments depend on the portfolio-allocation

rate ` and so does the bank’s expected final wealth E[WB(`, RD)].

Because of the bank’s limited liability its expected final wealth E[WB(`, RD)] simply

contains the expected difference of in- and outflows in those states where the bank

6.2. THE BANK’S OBJECTIVE 179

is able to pay back deposits and interest due as promised

E[WB(`, RD)

]=

∞∫

DRDD+WB

[ x · (D +WB)−D ·RD ] · 1

σ· ϕ(

x− µσ

) dx ,

simplifying to

E[WB(`, RD)

]= σ · (D +WB) · ϕ(

µ− DRDD+WB

σ)

+ [µ · (D +WB)−D ·RD] · Φ(µ− DRD

D+WB

σ) , (6.2.1)

where

Φ(µ− DRD

D+WB

σ) ≡ P[ x ≥ DRD

D +WB

] (6.2.2)

is the probability of deposits being fully redeemed. It can be aligned with the

probabilities qj given by (3.2.41) in the Bernoulli model. Henceforth, let

bsz =µ− DRD

D+WB

σ(6.2.3)

be the solvency barrier for the bank after having standardized the returns on the

loan portfolio by their expected return and their volatility. In contrast to the bank’s

success probability qj, j = 1, . . . , 4, that accounts for the lumped character of loan

redemptions and of full deposit redemption in the Bernoulli model, Φ(bsz) depends

on the loan portfolio risk characterized by `, the bank’s debt volume D, initial equity

WB, and hence leverage.

The difference µ · (D+WB)−D ·RD would be the bank’s expected final wealth if it

had unlimited liability. Intermediation is valuable to the bank as long as expected

wealth given by (6.2.1) exceeds the pure equity investment, i.e. maxµ1, µ2 ·WB.

If initial equity is zero, WB = 0, intermediation is even valuable for deposit interest

rates RD up to the expected return µ on the loan portfolio because of

σ ·D · ϕ(µ−RD

σ) > 0 .

The additional costs associated with the increasing risk of bankruptcy are given by

∂Φ(bsz)

∂D= − 1

σ· WB ·RD

(D +WB)2· ϕ(bsz) < 0 , (6.2.4)

i.e. by a decreasing probability of solvency.


The marginal return on the bank’s expected final wealth by each dollar of deposits

D raised is independent of theses costs as it is equal to

∂E[WB(·)]∂D

= σ · ϕ(bsz) + (µ−RD) · Φ(bsz)

+ σ · (D +WB) · ϕ(bsz) ·µ− DRD

D+WB

σ2· WBRD

(D +WB)2

︸︷︷︸≡−σ· ∂bsz

∂D

+ [µ(D +WB)−DRD] · ϕ(bsz) ·∂bsz∂D

= σ · ϕ(bsz) + (µ−RD) · Φ(bsz) . (6.2.5)

Thus the increasing risk of bankruptcy is not directly internalized by the risk-neutral

bank, but indirectly, via the deposits supply, by the risk-averse household. The

same argument as with respect to deposit supply applies Analogous to the portfolio-

allocation rate which can take effects in both directions.

6.3 The Household

As described in Section 3.2.3, the household can allocate its initial wealth, WH ,

between risky bank deposits, promising a pay-off of D · RD, and risk-free assets

yielding the exogenous gross return Rf . The household maximizes its expected

utility over final wealth WH resulting in its optimal supply of deposits. The

household is assumed to be risk-averse with a constant coefficient of absolute risk

aversion γ > 0. Its von Neumann-Morgenstern utility function over outcomes y is

given by

uH(y) = −e−γ·y .

Outcomes are given in dollars and thus, risk-aversion is measured in one over dollars

as utility is cardinal.

Since deposits are uninsured and are arranged by a standard debt contract, their

redemption depends on the performance of the bank’s loan portfolio and is thus

risky, featuring the following pay-off profile,

D = max min D ·RD, x · (D +WB) , 0 ,

where x is the stochastic, normally distributed gross return on the bank’s aggregate

6.3. THE HOUSEHOLD 181

loan portfolio according to (5.4.6).

The household judges deposit contracts according to their risk-return profiles.

The latter depend on the bank’s leverages, the exogenous risk-return profiles

of each sector-specific loan portfolio and the bank’s endogenous decisions about

the portfolio-allocation rate ` and the deposit interest rate RD. As there is no

asymmetric information, these inputs are perfectly observable to the household and

the bank credibly commits to each pair (`, RD). The household takes (`, RD) as

given when maximizing its expected utility for D.

The household’s final wealth amounts to

WH = max min D ·RD, x · (D +WB) , 0 + (WH −D) ·Rf ,

being concave in the deposit volume D for given portfolio returns x. Then the

household’s expected utility over final wealth from its portfolio decision is given by

E[uH(WH)

]=

0∫

−∞

−e−γ·(WH−D)·Rf · 1

σ· ϕ(

x− µσ

) dx

+

DRDD+WB∫

0

−e−γ[x(D+WB)+(WH−D)Rf ] · 1

σ· ϕ(

x− µσ

) dx

+

∞∫

DRDD+WB

−e−γ[DRD+(WH−D)Rf ] · 1

σ· ϕ(

x− µσ

) dx .

The first integral represents all states where the bank completely loses its loans.

Consequently, there are zero pay-offs from the deposit contract and only the risk-

free asset is redeemed with its interest bearing.

The second integral represents all states where the bank still goes bankrupt since it

cannot meet its obligations related to the debt contract, but where firms pay back

a positive amount on an aggregate level. Thus, the residual value from the loan

portfolio is transferred to the household in addition to the pay-off from the risk-free

asset.

The last integral summarizes all states where the bank can meet its obligations to

the household and the household is paid-off as promised.


Evaluating the integrals yields

E[uH(WH)

]= −e−γ(WH−D)Rf · Φ(−µ

σ)

−e−γ[µ(D+WB)+(WH−D)Rf− 12γσ2(D+WB)2]

·[

Φ(µ− γσ2(D +WB)

σ)− Φ(

µ− DRDD+WB

− γσ2(D +WB)

σ)

]

−e−γ[DRD+(WH−D)Rf ] · Φ(µ− DRD

D+WB

σ) .

(6.3.1)

The last and the first line are each given by the household’s utility of the respective

pay-offs weighted by the probability of the states considered. The intermediate lines

represent those states where the household is partially paid-off. On this domain,

expected utility depends on the well-known µ-σ representation of risky pay-offs that

exhibit normal returns and are judged according to CARA utility, adjusted by the

appropriate probability.

Since the household is only allowed to seize the bank’s assets in case of the bank’s

bankruptcy, this utility representation is multiplied by its respective probability. The

probability of these intermediate states are adjusted by the portfolio’s volatility σ

and the household’s coefficient of absolute risk-aversion γ.

The household’s supply of deposits arises from the following maximization problem:

maxD

E[uH(WH)

]s.t. 0 ≤ D ≤ WH . (6.3.2)

As in Section 3.2.3, the unconstrained optimization problem does not account

for the lower and the upper bound 0 ≤ D ≤ WH and will be referred to by

(6.3.2’). Unconstrained magnitudes will be super-indexed by u. The solution to

the Unconstrained Problem (6.3.2’) is called Du(`, RD).

Result 30. The household’s objective function E[uH(WH)] is strictly concave in D

for all D ∈ R. The unconstrained deposit supply Du(`, RD) that maximizes Problem

(6.3.2’) always exists and is unique.

Expected utility over final wealth is a strictly concave function of final wealth WH .

In turn, final wealth is concave in the deposit volume D for given portfolio returns.

The expectations operator preserves concavity. Thus, E[uH(WH)] is strictly concave

in D for all D ∈ R.

To establish the existence of a maximum Du, we consider the behavior of expected


Portfolio-allocation rate Deposit interest ra

te0.55

0.60

0.65

0.50

0.70 1.05

1.06

1.07

1.081.09

1.10

0.020.02

0.04

0.06

Dep

osit

inte

rest

rate

Portfolio-allocation rate

1.10

1.09

1.08

1.07

1.06

1.050.50 0.55 0.60 0.65 0.70

Figure 6.1: Illustration of Condition (6.3.4)

The figure illustrates the functional expression dependent on ` and RD that is assumed to be greaterthan zero according to Condition (6.3.4). It is shown for the parameter values from Table 6.1 giventhe relation D ≡ D. The graph on the left-hand side shows this functional expression dependent on` and RD. The graph on the right-hand side indicates that this functional expression approacheszero (and even becomes negative) if RD is close to Rf . Otherwise, the condition is satisfied for theparameters as given in Table 6.1.

utility for |D| approaching infinity. One can show that the expected utility function

diverges to minus infinity for both D −→ −∞ and D −→ ∞. In between,

expected utility is continuous. Thus, at least one D exists that maximizes the

household’s expected utility. Strict concavity of the expected utility function

establishes uniqueness. For details we refer to Appendix C.1.

Since Du(`, RD) may be negative or exceed WH , we define the constrained deposit-

supply function Ds(`, RD) analogous to (3.2.49):

Ds(`, RD) = min max Du(`, RD), 0 ,WH . (6.3.3)

The deposit-supply function can only be obtained numerically. For details of

calculations we refer to Section 6.6. Still, some basic general properties of the

deposit-supply function can be derived. First, let us assume the following relation

for the return on the household’s portfolio:

RD −Rf

σ− ϕ(bsz)

Φ(bsz)> 0, (6.3.4)


where

bsz ≡µ− DRD

D+WB

σwith D ≡ µ−Rf

γσ2−WB .

Technically, Condition (6.3.4) assures that the household’s first-order condition is

positive at D = D. Figure 6.2 illustrates this condition for the base case. Thus,

Du exceeds D, as shown in Appendix C.2. Economically, the household recognizes

the buffer function that the bank’s initial equity has concerning the redemption of

deposits. Conversely, if the household supplied D dollars to the bank and got back

(a contractually fixed fraction of) x ·(D+WB) in any state of the world, the amount

of deposits supplied would decrease with every dollar of equity initially provided in

a ratio of one to one, as given by (6.3.5), i.e. deposit supply would be equal to D

(adjusted by the appropriate parameter).

Unconstrained deposit supply is a smooth function in the portfolio-allocation rate

`, the deposit interest rate RD, and in all its parameters. It is affected by the trade-

off between the expected return on the loan portfolio and the associated volatility.

Likewise, a higher return Rf on the risk-free asset lowers deposit supply given the

deposit contract characterized by ` and RD. The following result summarizes these

properties:

Result 31. The deposit-supply function is continuous and the unconstrained deposit-

supply function is differentiable in the portfolio-allocation rate `, in the deposit

interest rate RD, and in all other parameters. The unconstrained deposit-supply

function strictly decreases in Rf .

If Approximation (5.4.14) is assumed and if Condition (6.3.4) is satisfied, the

unconstrained deposit-supply function exceeds

Du(`, RD) >µ−Rf

γσ2− WB . (6.3.5)

Moreover, the unconstrained deposit-supply function strictly increases in µ and

strictly decreases in σ. Let `max be the portfolio-allocation rate where Du(`, RD)

attains its maximum for given values of RD and let `σmin be the portfolio-allocation

rate that minimizes σ. Then, if `σmin ∈ (0, 1) holds, the following relations apply,

`max

> `σmin if µ1 > µ2

= `σmin if µ1 = µ2

< `σmin if µ1 < µ2

. (6.3.6)

Existence, given by Result 30, in conjunction with the implicit function theorem


leads to differentiability of the unconstrained solution (Mas-Colell/Winston/Green,

1995, p. 941f and Heuser, 1992, p. 295f). The constrained deposit supply Ds(`, RD),

given by (6.3.3), is differentiable in `, RD, and in all other parameters, except for

those parameter values where Ds(`, RD) equals zero or WH .

The behavior of the deposit-supply function in µ and σ is derived in Appendix C.2.

Since the deposit-supply function depends on ` solely via µ and σ, we can state for

the unconstrained deposit supply

∂Du

∂`=

∂Du

∂µ︸︷︷︸>0 Res. 31

·∂µ∂`

+∂Du

∂σ︸︷︷︸<0 Res. 31

·∂σ∂`

.

By assuming `σmin ∈ (0, 1), ∂σ∂`

changes signs on (0, 1) once in the variance minimum

portfolio-allocation rate. Suppose µ1 > µ2. Then ∂Dd

∂`> 0 holds for all ` ≤ `σmin .

Thus, the household demands more deposits as ` becomes higher. Above all, `max >

`σmin holds. Symmetrically, we can argue for µ1 < µ2 by starting the reasoning from

` ≥ `σmin .

Second, to characterize the deposit-supply function Ds(`, RD) with respect to the

deposit interest rate, it is again helpful to introduce the three critical deposit

interest rates RD(`), RD(`), and Ru

D(`) for every given portfolio-allocation rate `,

cf. Definitions (3.2.51) and (3.2.52).

RD(`) denotes the critical deposit interest rate where the unconstrained deposit-

supply function Du(`, RD) becomes zero:

RD(`) ∈ RD|RD = supRD

Du(`, RD) = 0 . (6.3.7)

Ru

D(`) is the deposit interest rate that maximizes the unconstrained deposit-supply

function Du(`, RD):

Ru

D(`) ∈RD|Du(`, R

u

D(`)) ≥ Du(`, RD) ∀ RD ≥ RD(`). (6.3.8)

RD(`) represents the minimum deposit interest rate that maximizes the

(constrained) deposit-supply function Ds(`, RD):

RD(`) = min

Ru

D(`), RD|RD = infRDDs(`, RD) = WH

, (6.3.9)

leading to the relation RD(`) ≤ RD(`) ≤ Ru

D(`) by definition if these maxima exist.


Portfolio-allocation rate Deposit interest ra

te0.55

0.60

0.65

0.50

0.70 1.05

1.06

1.07

1.081.09

1.10

0.020.02

0.04

0.06

Figure 6.2: Illustration of Condition (6.3.10)

This figure illustrates the dependence of the functional expression on ` and RD that is assumedto be lower than one according to Condition (6.3.10). The functional expression is shown forthe parameter values from Table 6.1 given the deposit-supply function Du(`, RD) which solves theMaximization Problem (6.3.2’). The condition is satisfied for the parameters as given in Table 6.1.

Moreover, we assumeϕ(busz)

Φ(busz)· D

u ·WB ·Rf

σ(Du +WB)2< 1 , (6.3.10)

where

busz ≡µ− Du·RD

Du+WB

σ,

to establish the following result which is in fact a re-formulation of Result 2

considering that we need not differentiate between different Cases as necessary in

the model discussed in Part II.

Result 32. If Condition (6.3.10) holds, exactly one interest rate RD(`) exists

for which the unconstrained deposit-supply function becomes zero, and exactly one

interest rate Ru

D(`) exists that maximizes the unconstrained deposit-supply function

with respect to RD. Furthermore, the constrained deposit-supply function Ds(`, RD)

increases strictly monotonically in RD for RD ∈ [RD(`), Ru

D(`)], and decreases

strictly monotonically on [Ru

D(`),∞).

The economic intuition for this result is the same as for Result 2: the risk-averse

investor with CARA utility will not increase its deposit supply beyond a given

threshold interest rate RD = RD(`), as the contingent pay-offs from the deposit

contract have already attained a given level of global satiation due to relatively

high values of RD. Moreover, the bank’s probability of solvency decreases for the

6.4. EQUILIBRIUM WITHOUT REGULATION 187

deposit volume D, as illustrated by (6.2.4).1 Technically, this proof follows the same

approach as that for Result 2 within the Bernoulli framework. Details are presented

in Appendix C.3. In fact, Ru

D(`) is implicitly given by

Ru

D(`) =(1 + γ ·Du ·Rf ) · Φ(b

u

sz)

ϕ(bu

sz) · Du·WB

σ·(Du+WB)2+ γ ·Du · Φ(b

u

sz)

(6.3.5)< Rf +

σ2

µ−Rf − γσ2 ·WB

,

with

bu

sz ≡µ− D

u·RuD(`)

Du

+WB

σ,

where Du ≡ Du(R

u

D(`), `) and where ` takes any arbitrary, but fixed value from

[0, 1].

The critical interest rate RD(`) barely exceeds Rf ,

RD(`) =1

Φ(µσ)·Rf = Rf + ε, ε > 0 , by (5.4.14) RD(`) ≈ Rf ,

i.e. the success probability of full deposit redemption approaches one as D goes

to zero. The economic reason is that an investment equal to zero dollars by the

household yields zero dollars with certainty at the end of the period. If final

payments deviated from zero, one party would always be granted an arbitrage

opportunity at the expense of the counter-party. Technically, this holds true because

of (5.4.14).

6.4 Equilibrium without Regulation

Without facing any regulation, the bank chooses the portfolio-allocation rate ` and

the deposit interest rate RD such that its Objective (6.2.1) given the household’s

deposit-supply function Ds(`, RD) is maximized:

max`, RD

E[WB(`, RD)

]

s.t. ` ∈ [0, 1] (6.4.1)

D ≡ Ds(`, RD) .

1In the model with Bernoulli distributed loan redemptions, the bank’s probability of solvencysolely depends on the Case j. The Case, in turn, is determined by the relation between D, RD,and `.


The pair (`∗, R∗D) denotes the maximizer to this problem and will be referred to

as the equilibrium without regulation. The basic properties, already shown and

discussed within the framework with the Bernoulli distribution, are valid too:

Result 33. An equilibrium (`∗, R∗D) always exists. The deposit interest rate, R∗D,

satisfies R∗D ∈ (RD(`), RD(`)]. Hence, the optimal deposit volume D∗ ≡ Ds(`∗, R∗D)

is always strictly positive.

Expected final wealth E[WB(`, RD)] is continuous in ` and RD. The portfolio-

allocation rate ` is restricted to the compact set [0, 1] by definition. The domain of

RD can be restricted to a compact set as well without excluding any optima. Since

∂E[WB(·)]∂RD

= − D · Φ(bsz) < 0

holds for fixed D, the bank does not increase deposit interest rates beyond RD(`):

for RD higher than RD(`), expected wealth decreases by both higher interest rates

RD and lower deposit volumes D, where the latter is due to Result 32. As for

all RD < RD(`) the deposit volume remains zero, deposit interest rates below

RD(`) do not alter the value of the bank’s expected final wealth and can thus be

skipped. Consequently, R∗D must be restricted to [RD(`), RD(`)] and a solution to

the Maximization Problem (6.4.1) always exists. Because of

dE[WB(·)]dRD

∣∣∣∣∣RD=RD

=

(∂E[WB(·)]

∂D· ∂D

u(·)∂RD

)∣∣∣∣∣RD=RD

+∂E[WB(·)]∂RD

∣∣∣∣∣RD=RD

(6.2.5)= − Du(`, RD) · Φ(bsz)

+ σ · ϕ(bsz) + (µ−RD) · Φ(bsz) ·∂Du(·)∂RD

∣∣∣∣RD=RD

=σ · ϕ(bsz) + (µ−RD) · Φ(bsz)

· ∂D

u(·)∂RD

∣∣∣∣RD=RD

> 0 ,

(6.4.2)

it is not optimal for the bank to forgo debt finance, i.e. D = 0 is never optimal.

Consequently, R∗D lies in (RD(`), RD(`)] and D∗ is strictly positive. Note that R∗D =

RD(`) can be an equilibrium if RD(`) < Ru

D(`) holds; that is, whenever the household

is constrained by its initial wealth in equilibrium, Ds(`∗, RD(`)) ≡ WH .

If the household is not constrained by its initial wealth, choosing R∗D = RD(`) cannot

6.4. EQUILIBRIUM WITHOUT REGULATION 189

be optimal for the bank because of

∂E[WB(·)]∂RD

∣∣∣∣∣RD=RD(`)

= − Du(`, RD) · Φ(bsz)


∣∣∣∣RD=RD(`)

= − Du(`, RD(`)) · Φ(bsz) < 0 .

The risk-averse household affects the bank’s choice of the optimal portfolio-allocation

rate such that the risk neutral bank management does not allocate all its funds to

the loan portfolio with the higher expected return. Still, the relation between both

loan portfolios’ expected returns affects the bank’s decision stronger than it does

with the household’s:

Result 34. Assume that `∗ ∈ (0, 1), `σmin ∈ (0, 1), and that the household is not

constrained by its initial wealth. Then the following relations hold for the optimal

portfolio-allocation rate:

`∗

> `max > `σmin if µ1 > µ2

= `max = `σmin if µ1 = µ2

< `max < `σmin if µ1 < µ2

. (6.4.3)

The derivation is as follows. By the conditions set out in the result, no further

constraints must be considered when analyzing the first-order constraints. The

first-order condition to (6.5.3) with respect to RD is given by

∂E[WB(·)]∂RD

= − Du(`, RD) · Φ(bsz)


!= 0 .

Thus,∂E[WB(·)]

∂D= σ · ϕ(bsz) + (µ−RD) · Φ(bsz) > 0 (6.4.4)


holds. The first-order condition with respect to ` reads

∂E[WB(·)]∂`

= [ Du(`, RD) +WB ] ·ϕ(bsz) ·

∂σ

∂`+ (µ1 − µ2) · Φ(bsz)

︸︷︷︸≡E[WB ]`

+ σ · ϕ(bsz) + (µ−RD) · Φ(bsz)︸︷︷︸≡E[WB ]D

· ∂Du(·)∂`

!= 0 .

Since Du(`, RD) +WB > 0 and E[WB]D > 0 hold, E[WB]` and ∂Du(·)∂`

are either of

opposite sign or zero.

If µ1 = µ2 the first-order condition simplifies such that E[WB]` and ∂Du

∂àre reduced

to multiples of ∂σ∂`

. One solution is then `∗ = `σmin . Consequently, the household’s

deposit supply is maximized with respect to ` due to Result 31. As E[WB]D > 0

holds, this is in turn in the interest of the bank. Hence, the optimal portfolio-

allocation rate is given by `∗ = `σmin and is unique. Consequently,

σ · ϕ(bsz) + (µ−RD) · Φ(bsz) > [Du(`, RD) +WB] · ϕ(bsz) (6.4.5)

must hold, as ∂2E[WB ]˜2 < 0 is unambiguously linked to a unique maximum in `.

Consider now µ1 > µ2: there is no incentive for the bank to put a weight on the first

sector less that is lower than the variance-minimum loan-allocation rate. If it did, it

could improve in two ways by raising the portfolio-allocation rate: first, a higher `

would directly increase the bank’s expected final wealth. Second, a higher ` would

raise deposits D according to Result 31 and thus increase the bank’s expected final

wealth as well.

The first (direct) effect is formally due to the assumption sgn(E[WB]`) = sgn(µ1 −µ2). By this property, E[WB]` > 0 holds and thus ∂Du

∂`< 0. By the latter we can

conclude `∗ > `max.

For µ1 < µ2 we can argue analogously.

6.5. EQUILIBRIUM WITH REGULATION BY A VALUE-AT-RISK APPROACH 191

6.5 Equilibrium with Regulation by a Value-at-

Risk Approach

The bank faces the same sort of VaR regulation as described in Section 3.3.3.1:

regulation requires the bank that unexpected losses,

[E[WB] − L

]+

,

must not exceed a given threshold (a fraction/multiple τ of initial bank equity WB)

by more than a given probability p:

P(

E(L)− L ≥ τWB

)≤ p . (6.5.1)

Since the gross return on the whole loan portfolio is normally distributed, the VaR

requirement results in the following upper bound on the deposit volume2

D ≤ w(`) ·WB where w(`) = max − τ

Φ−1(p) · σ − 1 , 0 , p <1

2. (6.5.2)

The regulatory constraint becomes weaker as the bank gains more equity or the

value of the parameter τ increases. The former applies to p > Φ(− τσ) and p < 1

2, for

the latter to hold, only p < 12

must be fulfilled. The regulatory constraint strictly

increases in p for either p ∈ (0, 12) or p ∈ (1

2, 1).

As volatility is the only parameter characterizing the distribution of the loan

redemptions that goes into (6.5.2), the VaR constraint seems to foster decisions that

reduce risk. The leeway for the deposit volume is the highest under regulation if the

bank chooses the variance minimum portfolio-allocation rate. Increasing portfolio-

allocation rates that are below `σmin imply a higher feasible deposit volume, and vice

versa for portfolio-allocation rates that are above `σmin ,

∂w

∂`=

τ

Φ−1(p) · σ ·∂σ

∂`

> 0 if ` < `σmin

= 0 if ` = `σmin

< 0 if ` > `σmin

.

However, under the conditions of Result 34, the equilibrium portfolio-allocation

rate cannot be further or differently characterized than done under the equilibrium

without regulation.

2Cf. Dangl/Lehar (2004, p. 108) for a similar VaR based formulation of capital requirementsconsidering normally distributed portfolio returns.


Sufficiently high volatilities σ1 and σ2 may prevent the regulated bank from issuing

deposits. More precisely, if σ ≥ − 1Φ−1(p)

holds for any feasible portfolio-allocation

rate `, the bank is only able to allocate its equity WB to that loan portfolio which

yields the higher expected return µi.

The bank’s maximization problem under regulation is given by

max`, RD

E[WB(`, RD)

]

s.t. ` ∈ [0, 1] (6.5.3)

D ≡ Ds(`, RD)

D ≤ w(`) ·WB .

The tuple (`V , RVD) denotes the equilibrium under the VaR regulation. Regulation is

said to be binding, if the laissez-faire equilibrium (`∗, R∗D) does not fulfill the given

level of confidence p. The properties of the equilibrium without regulation presented

in Results 33 and 34 carry over to the equilibrium under regulation. Apart from

lower total loan and deposit volumes, no further properties of both equilibria can

be shown in general.

Result 35. An equilibrium (`V , RVD) always exists. Assume that `V ∈ (0, 1), `σmin ∈

(0, 1), and that the household is not constrained by its initial wealth. Then the

optimal portfolio-allocation rate `V also satisfies the relations in (6.4.3).

The argument for the existence of (`V , RVD) runs as in the unregulated case. The

additional regulatory Condition (6.5.2) restricts the space of the choice variables

(`, RD) further. The argument for the behavior of the optimal portfolio-allocation

rate `V can be shown analogously as follows. Consider the Lagrangian3

L(`, RD, λ) = E[WB(`, RD)

]− λ · [Du(`, RD)− w(`) ·WB] .

The first-order condition with respect to RD becomes4

∂L(·)∂RD

= − Du(`, RD) · Φ(bsz)

+ σ · ϕ(bsz) + (µ−RD) · Φ(bsz)− λ ·∂Du(·)∂RD

!= 0 ,

3Note that assuming `V ∈ (0, 1) is equivalent to assuming σ < 1− Φ−1(p) as a violation of the

latter inequality implies either `V = 0 or `V = 1, dependent on what loan portfolio yields thehigher return, and zero deposits. Moreover, w(`) becomes differentiable.

4Cf. the derivation of Formula (6.4.2).

6.5. EQUILIBRIUM WITH REGULATION BY A VALUE-AT-RISK APPROACH 193

resulting in

σ · ϕ(bsz) + (µ−RD) · Φ(bsz)− λ > 0 .

Thus, the marginal impact of deposits is also strictly positive under regulation and

can be more concisely written as LD ≡ E[WB]D − λ > 0. The first-order condition

with respect to ` reads5

∂L(·)∂`

= [Du(`, RD) +WB] ·ϕ(bsz) ·

∂σ

∂`+λ

σ· ∂σ∂`

+ (µ1 − µ2) · Φ(bsz)

︸︷︷︸L`

+ σ · ϕ(bsz) + (µ−RD) · Φ(bsz)− λ ·∂Du(·)∂`

!= 0 .

Due to LD > 0 and Du(`, RD) +WB > 0, the derivative ∂Du(·)∂`

and the remainder

L` are either of opposite sign or zero.

For µ1 = µ2 we can argue exactly as we did for Result 34. Moreover, choosing

`V = `σmin maximizes the eligible deposit volume according to (6.5.2).

Again, for µ1 > µ2, there is no incentive for the bank to put a weight on the first

sector that is lower than the variance-minimum loan-allocation rate. If it did, it could

now improve even in three ways by raising the portfolio-allocation rate: besides the

former two reasons that have already been mentioned in connection with Result 34,

increasing ` toward `σmin also increases deposit volume that is eligible by regulation.

Thus, `V > `max > `σmin is obtained. We can argue symmetrically for µ1 < µ2.

If loan redemptions from both loan portfolios are equal in their expected returns,

µ1 = µ2, as well as in their volatilities, σ1 = σ2, and if regulation is binding, the

equilibrium deposit volume DV equals

DV ≡ Du(1

2, RV

D) =

(− τ ·

√2

Φ−1(p) · σ1 ·√

1 + ρ− 1

)·WB, Φ(− τ

σ) < p <

1

2.

In particular, the equilibrium total loan/deposit volume strictly decreases in the

single portfolios’ volatilities σ1 and in their correlation ρ and remains unaffected by

changes in expected returns. If the bank is regulated by fixed risk weights, as studied

in Section 3.3.2 for Bernoulli-distributed loan redemptions, the total loan volume

under regulation is simply given by LS = 1c·c1 ·WB. As long as risk is measured by

the volatility σi, the VaR approach affects the total loan volume in a pro-cyclical

way compared to the total loan volume granted under a fixed-risk weight regime.

5The expression λ · τ−Φ−1(p)σ2 ·WB · ∂σ∂` has been substituted by [Du(·) +WB ] · λσ · ∂σ∂` because

the Regulatory Constraint (6.5.2) holds with equality in equilibrium.


Table 6.1: Parameter values of the base case, normal model

This table reports the base case parameter values used for the numerical analysis of the model setforth in this chapter. The parameter values of the bank’s initial wealth, the household’s coefficientof absolute risk aversion, and the gross return rate on the risk-free asset are identical to those givenby Table 4.1. The parameter values of the expected gross returns and the return volatilities, andthe inter-sectoral loan-return correlation result from the mixture model set forth in Chapter 5 withthe parametrization according to Table 5.1. Expectations are precisely quoted whereas volatilitiesand the correlation are rounded to six leading digits.

WB γ Rf τ µ1 µ2 σ1 σ2 ρ

100 0.008 1.05 1 = 1.14425 = 1.188 ≈ 0.0811463 ≈ 0.119495 ≈ 0.000426955

However, once it is recognized that both the portfolio volatility σi as well as the risk

weight ci depend positively on the default probability 1−pi, 1−pi ∈ (0, 12), it is still

unclear whether LS or LV reacts stronger to changes in 1− pi. Arguing strictly by

the Standardized Approach on the one hand and the internal-ratings-based approach

on the other, LS is piecewise insensitive with up to three jumps according to Table

3.4 whereas LV changes smoothly.

6.6 Numerical Analysis of Regulatory Impacts

Let us analyze this model with the parameter values as given in Table 5.1 which

result in the same values of the redemption distributions on the individual firm

loan level as in Chapter 4. Because of the inner-sectoral correlations between firms,

sector-wise volatilities σi depart slightly from the values shown in Table 4.1. The

Bernoulli mixture model set forth in Chapter 5 given the parametrization according

to Table 5.1 leads, however, to a cross-sectoral correlation that is much lower

than the correlation in the Bernoulli model. Table 6.1 shows these values. For

simplicity, comparative statics are performed as if the household’s initial wealth WH

is arbitrarily high, such that only unconstrained equilibria are considered under the

base case. The regulatory multiplier τ is set equal to one, reflecting the notion of

the IRB approach to determine the overall loan portfolio’s VaR. For a discussion on

the meaning of τ , we refer to p. 96.

The deposit-supply function is computed as follows: first the household’s expected

utility is numerically maximized for given tuples (`, RD). Second, by linearly

interpolating the triples (`, RD, D), we can set up the household’s deposit supply

6.6. NUMERICAL ANALYSIS OF REGULATORY IMPACTS 195

Table 6.2: Alternative parameter values, normal model

The upper and the lower panel report an alternative parametrization each. ∆` and ∆RD representthe step-size of the grid in the `−RD space on which the household’s expected utility is maximizedwith respect to deposit supply D .

Panel A:

µ1 µ2 σ1 σ2 ρ WB γ WH Rf τ p ∆` ∆RD1.06 1.08 0.02 0.03 0 102 0.4 4000 1 1 0.1% 0.001 0.00001

Panel B:

µ1 µ2 σ1 σ2 ρ WB γ WH Rf τ p ∆` ∆RD1.08 1.12 0.06 0.1 0.3 100 0.2 4000 1 1

3 1.0% 0.01 0.0001

function Ds(`, RD) which is cut off at 0. We can then numerically determine the

bank’s maximum choice both with and without regulation. The step lengths and

the domains of ` and RD may vary with the (regulatory) regime considered when

equilibria were computed under the base case parametrization given by Table 6.1.

Beyond that base case we also consider two further parameter constellations to

support our exemplary findings (Buhler/Koziol/Sygusch, 2007). The first set of

parameters is given by Table 6.2, Panel A. These values are chosen such that at

WB = 102 regulation ceases to bind (if we consider a step size of one concerning

WB). As regulation, only a VaR approach is considered. The level of confidence

is set equal to p = 0.1% and the scaling parameter τ is equal to one. The bank’s

assets are riskier in the scenario characterized by the parameter values shown in

Table 6.2, Panel B. The VaR approach considered is tighter with respect to the

scaling parameter τ but weaker in terms of the confidence level p. The coefficient

of absolute risk-aversion γ has been set to a lower level to allow for a propensity

to supply deposits that is high enough in order to make the regulatory constraint

binding.

6.6.1 Equity Shocks

Figure 6.3 shows the (equilibrium) total loan volumes with and without regulation

as a function of the bank’s initial capital WB. The total loan volume increases


0

005

0001

0051

0002

0052

0003

0053

0004

0054

0005

0055

0006

0056

0007

0057

0008


Tota

l loa

n vo

lum

enoitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa


This figure illustrates the dependence of the equilibrium total loan volume L = L1 + L2 on thebank’s initial equity WB. The total loan volume is shown for five different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with confidence levels p = 1.0% and p = 0.1%.

monotonically in the bank’s initial equity under all regimes. Without regulation,

the bank demands and the households supplies deposits even if the bank owners do

not initially bring in any equity. This is not feasible under the capital requirements

considered. In contrast to the preceding Bernoulli-distribution framework, the

VaR approach requires a strictly positive amount of initial equity for debt-financed

lending due to the Regulatory Constraint (6.5.2) that arises out of the normality

of aggregate loan redemptions. Hence, regulation is binding at least for values of

initial equity around zero. As all regulatory regimes link the feasible total loan

volume to the bank’s initial equity, the former will strictly increase in WB, at least

on average, or if countervailing indirect effects via the loan-allocation rate or the

deposit interest rate can be neglected. If there is some point WB where regulation

ceases to bind, the regulated total loan volume will increase more steeply on average

than the unregulated volume in order to catch up with the latter. As a consequence,

regulation has a pro-cyclical effect on average as already confirmed within the model

in Chapter 4.

This pro-cyclicality on average can be observed in connection with the Standardized

Approach. The slopes, both under equal risk weights and under different risk

weights, exceed the slope of the unregulated total loan volume. The latter exceeds,


%0.5

%2.5

%4.5

%6.5

%8.5

%0.6

%2.6

%4.6

%6.6

%8.6


Dep

osit

inte

rest

rate

%0000.5

%4000.5

%8000.5

%2100.5

%6100.5

%0200.5

%4200.5

%8200.5

Dep

osit

inte

rest

rate

(VaR

app

roac

hes)

o/w noitaluger

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa

Figure 6.4: Deposit interest rate as a function of the bank’s initial equity

This figure illustrates the dependence of the equilibrium deposit interest rate RD on the bank’s initialequity WB. Deposit interest rates are shown for five different regimes: the laissez-faire equilibrium,the Standardized Approach with both equal and different risk weights, and the VaR approach withconfidence levels p = 1.0% and p = 0.1%.Left-hand scale: deposit interest rates granted without regulation and under the StandardizedApproach.Right-hand scale: deposit interest rates granted under the VaR approach.We note that the deposit interest rate under the VaR approach with p = 0.1% barely exceeds therisk-free rate Rf − 1 = 5.0%, i.e. ranges between 5.0000124% and 5.0000174% for WB ∈ [0, 600].

however, the slopes of the total loan volumes under both VaR approaches which is

here due to the low regulatory parameter τ , τ = 1.

Figure 6.4 shows the deposit interest rates under the regimes considered. As in

the Bernoulli framework, the deposit interest rate does not only incorporate the

bank’s credit risk, but serves also as the instrument to fix the total loan and deposit

volume. Consequently, the deposit interest rates under regulation are lower than

their unregulated counterparts. Furthermore, deposit interest rates strictly increase

as a function of the bank’s capital under the VaR approaches.

This behavior can also be observed under the Standardized Approach with fixed

risk weights for lower values of initial equity. But thereafter, the risk-reducing

effect dominates and the deposit interest rates shrink as a result of increased

diversification: the kink in the deposit interest rate under the regime with equal

risk weights is associated with the change to allocating strictly positive amounts of

capital as loans to firms from the first sector. This kink in deposit interest rates is


%0

%01

%02

%03

%04

%05

%06

%07

%08

%09

%001

006045084024063003042081021060

ytiuqelaitinis'knaB

Por

tfolio

allo

catio

n ra

teo/w noitaluger

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa

Figure 6.5: Portfolio-allocation rate as a function of the bank’s initialequity

This figure illustrates the dependence of the equilibrium portfolio-allocation rate ` on the bank’sinitial equity WB. Portfolio-allocation rates are shown for five different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with confidence levels p = 1.0% and p = 0.1%.

reflected by the raise of the portfolio-allocation rate from zero to strictly positive

levels from WB ≈ 93 on in Figure 6.5. For c1 = 0.5, c2 = 1, this kinks already occurs

at WB ≈ 50.

The effects of higher risk-taking if risk weights are equal also arise here. Employing

different risk weights, c1 < c2, results in high allocation rates for loans to the first

sector if the bank’s initial wealth is low (Fig. 6.5). By doing so, the bank can attract

deposits relatively cheaply from the household. In particular, the return volatilities

on the loan redemptions, σ, and the return volatilities on deposits, σD, are lower

than if loans from both sectors are equally weighted (cf. Fig. 6.6), whereas σD is

analogously defined to (4.2.1) as

σD ≡√

V( DD

) . (6.6.1)

The risk-reducing effect of unequal risk weights is also reflected by the probability

of full deposit redemption which mostly exceeds that under equal risk weighting

(cf. Fig. 6.7).

Figure 6.6 also exemplifies the cushion effect of the bank’s initial equity which has


00.0

20.0

40.0

60.0

80.0

01.0

21.0


Ret

urn

vola

tiliti

es (l

oans

)

000.0

500.0

010.0

510.0

020.0

520.0

Ret

urn

vola

tiliti

es (d

epos

its)


dezidradnatshtiwhcaorppa0.1=2c,5.0=1c

)snaol(


0.1=2c=1c)snaol(


dezidradnatshtiwhcaorppa0.1=2c,5.0=1c

)stisoped(


0.1=2c=1c)stisoped(


This figure illustrates the dependence of the equilibrium return volatilities on the bank’s initial equityWB. Solid and dashed lines and dots, respectively, represent the volatilities of returns on deposits,σD. Diamonds and crosses represent the volatilities of returns on total loans, σ. Return volatilitiesare shown for five different regimes: the laissez-faire equilibrium, the Standardized Approach withboth equal and different risk weights, and the VaR approach with confidence levels p = 1.0% andp = 0.1%.

been already confirmed for the Bernoulli model in Part II.6 This cushion effect can

be observed for all regimes by the gap between volatilities of returns on the whole

loan portfolio and the volatilities of deposit returns. Without regulation, this gap

strictly increases with increasing WB and is even positive for WB = 0 since the

maximum pay-off from deposits is fixed and bounded. Note that this initial gap

also arises if repayments from the loan portfolio are bounded from above as it is the

case in the framework with a Bernoulli distribution.

Figure 6.8 illustrates that the bank may prefer granting loans to firms of the first

sector compared to the non-regulated case, if it can weight the two types of loans

differently. Compared to the regime with equal risk weights, different risk weights

6Result 4 shows the cushion effect to exist in Cases 2 and 3 out-of-equilibrium whereas Result10 refers to the total cushion effect under Case 4 in equilibrium and Result 12 to the partial cushioneffect in equilibrium if both projects are equal. Figures 4.8 and 4.9 illustrate the return volatilitieswithin the Bernoulli model for its base case parametrization.


19.0

29.0

39.0

49.0

59.0

69.0

79.0

89.0

99.0

00.1

10.1


Pro

babi

lity

of fu

ll de

posi

t red

empt

ion

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa

Figure 6.7: Probability of full deposit redemption

This figure illustrates the dependence of the equilibrium portfolio-allocation rate ` on the bank’sinitial equity WB. Portfolio-allocation rates are shown for five different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with confidence levels p = 1.0% and p = 0.1%.

may result in a higher sensitivity of the total loan volume to shocks in WB as the

(average) risk weight of the loan portfolio is lower than the (average) risk weight

under the regime with flat weights equal to one, cf. Figures 6.3 and 6.8. Considering

the total loan volume, the sensitivity ranges between 16.38 and 25 with c1 = 0.5

and c2 = 1 and is thus up to twice as much as the sensitivity with weights equal

to c1 = c2 = 1. This examples thus illustrates that more risk-sensitive capital

requirements may result in pro-cyclicality. The VaR approach with τ = 1 leads to

the opposite conclusion for the range of values of WB considered. That is, formally

w(`V )︸︷︷︸direct effect

+ w′(`V ) · d`V

dWB

·WB

︸︷︷︸indirect effect

< k(`S)︸︷︷︸direct effect

+ k′(`S) · d`S

dWB

·WB

︸︷︷︸indirect effect

holds. For most of the values of WB considered, the indirect effect is larger under

the Standardized Approach, regardless of the risk weighting, than under any of

both VaR approaches (cf. Fig. 6.5). The same applies to the direct effects, too. In

particular, w(`V ) < k(`S) is fulfilled if τ does not exceed

τ <[−Φ−1(p)

]· σ|`=`V ·

1

c · [c1`S + c2(1− `S)],


0

005

0001

0051

0002

0052

0003

0053

0004

0054


Loan

por

tfolio

1

noitalugero/w


1=2c=1c


1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

Figure 6.8: Loan volume to Sector 1 as a function of the bank’s initialequity

This figure illustrates the dependence of the equilibrium loan volume granted to Sector 1 on thebank’s initial equity WB. Loan volumes L1 are shown for five different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with confidence levels p = 1.0% and p = 0.1%.

where[−Φ−1(p)] ≥ 2.32635 if p ≤ 0.01 ,

σ ∈[ √

(1−ρ2)σ21σ

22

σ21−2ρσ1σ2+σ2

2,maxσ1, σ2

], and

1c·[c1`S+c2(1−`S)]

∈ [813, 62.5] if 0.2 ≤ ci ≤ 1.5 .

Hence, the lowest upper bound for τ such that the direct effect is smaller under the

VaR approach than under the Standardized Approach is given by

τ < 19.3862 ·√

(1− ρ2)σ21σ

22

σ21 − 2ρσ1σ2 + σ2

2

,

amounting to 1.3016 in the base case as given by Table 6.1, to 0.322607 for the

parameters as given by Table 6.2, Panel A, and to 1.1096 if Panel B is considered.

Given the parametrization according to Table 6.2, Panel A, the VaR-regulated total

loan volume actually increases on average by 18.260 if it is considered as a function

of WB (cf. Fig. 6.9), but only by roughly 1.4 if the parameters from Panel B are

considered (cf. Fig. 6.10).

Within the presented framework, all forms of regulation enhance the depositor’s


protection in terms of the probability of full redemption of deposits (including

interest, cf. Definition (6.2.2)), as shown by Figure 6.7. The portfolio-allocation

rates in Figure 6.5 indicate that volatilities of returns on the loan portfolio and also

on deposits are higher under both forms with fixed risk weights ci than if the bank

is unregulated. This also holds true for the regime with different risk weights as

the scope for diversification is not fully exploited either. Only the VaR approaches

effectively dampen the return volatilities because of the low deposit volumes and

the low deposit interest rates on the one hand and the negligible impact on the

portfolio-allocation rate on the other. Thus, a reduction in leverage already leads

to a reduction of the probability of full deposit redemption.

Next, let us consider the parameter values, as shown in Table 6.2, Panel A and B. The

respective VaR approaches result in similar patterns concerning total loan volumes.

Figure 6.9 illustrates that the unregulated total loan volume may be more sensitive to

changes in the bank’s initial equity than the regulated volume for low equity values.

The curve of total loan volumes is clearly concave such that it is emphasized once

more that pro-cyclicality of regulation with respect to changes in initial equity may

only hold on average, but not at every point. The total loan volumes emerging from

the parameter values from Table 6.2, Panel B, further confirm this point, cf. Figure

6.10.

To sum up, these numerical studies suggest that regulation enhances pro-cyclicality

in bank lending in general if regulation is binding and a shock has occurred that has

impaired the bank’s initial equity. In particular, this notion applies where regulation

has just started to bind: let WB be the level of equity below which regulation is

binding. Once the bank falls below the critical value WB, we obtain Lr < L∗ by

definition and thus we should observe ∆Lr

∆WB> ∆L∗

∆WB> 0 for some WB still sufficiently

close to WB.

However, examples have also shown that the sensitivity of total lending may even be

higher without regulation for low values of initial equity WB. The lower is WB, the

more pronounced the cushion effect becomes that equity signals to the risk-averse

depositors. Thus, the total loan/deposit volume that can be raised becomes more

sensitive to changes in WB, the closer the values considered for WB are to zero. In

contrast, the sensitivities of total loan volumes under either sort of regulation are

dominated by the almost linear relation between total lending and initial equity.


0

002

004

006

008

0001

0021

0041

0061

0081

0002

0022

0042

0062


Tota

l loa

n vo

lum

enoitalugero/w

ehttaRaVlevel%1.0

Figure 6.9: Total loan volume as a function of WB, Table 6.2, Panel A

This figure illustrates the dependence of the equilibrium total loan volume L on the bank’s initialequity WB. Results are based on the parametrization given by Table 6.2, Panel A. Total loanvolumes L are shown for two different regimes: the laissez-faire equilibrium and the VaR approachwith confidence level p = 0.1%.

0

002

004

006

008

0001

0021

0041

0061


Tota

l loa

n vo

lum

e

noitalugero/w

ehttaRaVlevel%0.1

Figure 6.10: Total loan volume as a function of WB, Table 6.2, Panel B

This figure illustrates the dependence of the equilibrium total loan volume L on the bank’s initialequity WB. Results are based on the parametrization given by Table 6.2, Panel B. Total loanvolumes L are shown for two different regimes: the laissez-faire equilibrium and the VaR approachwith confidence level p = 1.0%.


0

005

0001

0051

0002

0052

0003

0053

0004

0054

0005

0055

81.171.161.151.141.131.121.111.101.190.180.170.160.150.1rotceS snaolnonruterdetcepxE

Tota

l loa

n vo

lum

e

noitalugero/w


1=2c=1c


1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

1

Figure 6.11: Total loan volume as a function of µ1

This figure illustrates the dependence of the equilibrium total loan volume L on the expected grossreturn of first-sector loans, µ1. Total loan volumes L are shown for five different regimes: thelaissez-faire equilibrium, the Standardized Approach with both equal and different risk weights, andthe VaR approach with confidence levels p = 1.0% and with p = 0.1%.

6.6.2 Expected Return Shocks

Figure 6.11 shows the comparative static results for expected return shocks

concerning redemptions from the first loan portfolio. The higher the expected return,

the more loans are granted to that sector (not shown). Even the total loan volume

strictly increases under all regimes. The latter need not be the case, as Figure 6.12

highlights.

As regulation puts an upper bound on the feasible total loan/deposit volume by

the Standardized Approach according to (3.3.21) and the VaR approach according

to (6.5.2), respectively, and as the unregulated total loan volume strictly increases,

regulation has in general a dampening effect on the loan volume sensitivities to

expected return shocks. Under risk-adjusted portfolio weights with c1 = 0.5 and

c1 = 1, however, sensitivities may be higher which is mainly due to changes from

one extreme portfolio allocation to the other. In the base case, if expected returns

on the first portfolio are considerably low (µ1 ≤ 1.09390), the bank only grants

loans to firms of Sector 2. Likewise, if µ1 ≥ 1.16485, solely firms of the first sector

can borrow funds. In between, there is a relatively sharp increase in the portfolio-

allocation rate and thus in the aggregate loan volume to Sector 1 firms. Because


008

0001

0021

0041

0061

0081

0002

0022

0042

0062

0082

0003

0023

81.161.141.121.101.180.160.140.120.100.1nonruterdetcepxE

Tota

l loa

n vo

lum

e

noitalugero/w

ehttaRaVlevel%1.0

noitalugero/w

ehttaRaVlevel%1.0

rotceS snaol1

Figure 6.12: Total loan volume as a function of µ1, Table 6.2, Panel A

This figure illustrates the dependence of the equilibrium total loan volume L on the expectedgross return of first-sector loans, µ1. Results are based on the parametrization given by Table6.2, Panel A. Total loan volumes L are shown for two different regimes: the laissez-faire equilibriumand the VaR approach with confidence level p = 0.1%.

of augmented expectations, the total loan volume increases strongly as well. The

sensitivity of the regulated total loan volume may outreach the sensitivity of the

unregulated total loan volume as indicated by Figure 6.11. The same applies to L1

whereas the aggregate loan volume granted to Sector 2 is sloped downwards and

thus behaves counter-cyclically.

With parameter values according to Table 6.2, Panel A, the portfolio-allocation rate

strictly increases under both regimes and does so even further for µ1 > µ2 after

they have crossed the variance-minimum portfolio-allocation rate `σmin , where `σmin

is chosen if µ1 attains µ2 = 1.08 (cf. Result 35)7 Hence, the maximum feasible total

loan volume under the VaR approach reaches its global maximum value over all

expected returns, as the variance is the lowest at this point (cf. the VaR Constraint

(6.5.2)). As σ strictly increases again, if µ1 increases beyond µ2, a decreasing total

loan volume is obtained under VaR-based regulation due to the Constraint Function

(6.5.2).

7The comparative static analysis in µ1 under the base case has been performed until µ1 reachesµ2. This example is meant to show that the VaR-regulated total loan volume becomes counter-cyclical if µ1 increases beyond µ2. This phenomenon could be observed under the base case aswell.


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30.0

40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0

81.171.161.151.141.131.121.111.101.190.180.170.160.150.1

nonruterdetcepxE

Ret

urn

vola

tiliti

es



,5.0=1c 0.1=2c)stisoped(

ehttaRaVlevel%1.0

)stisoped(



,5.0=1c 0.1=2c)snaol(

ehttaRaVlevel%1.0

)snaol(rotceS snaol1

Figure 6.13: Return volatilities as a function of µ1

This figure illustrates the dependence of the equilibrium return volatilities on the expected grossreturn of first-sector loans, µ1. Solid and dashed lines represent volatilities of returns on deposits,σD. Diamonds and crosses represent volatilities on total loans, σ. Return volatilities are shownfor three different regimes: the laissez-faire equilibrium, the Standardized Approach with differentrisk weights, where c1 = 0.5 and c2 = 1, and the VaR approach with confidence level p = 0.1%.

Figure 6.13 illustrates the issue of risk-taking beyond the choice of portfolio-

allocation rates. Evidently, the extreme allocations under the Standardized

Approach with different risk weights translate into higher return volatilities

concerning both the loan portfolio and the deposits. As a consequence, risk-sensitive

weights do not mitigate volatilities compared to the regime with fixed risk weights

except for a rather narrow range of intermediate values.8 At µ ≈ 1.125, the regulated

bank chooses the variance-minimum loan-allocation rate though µ1 < µ2 = 1.188

still holds. With increasing µ1, the bank prefers raising expected returns at the

cost of increasing risk and the return volatility starts to rise until σ = σ1 is reached

by `S = 100% from roughly µ1 = 1.16714 on. In contrast, the unregulated loan-

allocation rate `∗ attains the variance-minimum at µ1 = µ2 (not shown). The latter

applies to the VaR-regime as well.

The probability of full deposit redemption is highest under the VaR approaches,

namely throughout higher than 99.98%. Under the Standardized Approach,

8If c1 = c2 = 1 holds, the loan-portfolio return volatility σ is equal to 0.11949 for µ1 < 1.13738.It strictly decreases for µ1 > 1.13738, and achieves its minimum equal to 0.06745 in µ1 = 1.18773.The deposit return volatility is similar to that under the regime with different risk weights.


0

006

0021

0081

0042

0003

0063

0024

0084

0045

0006

21.011.001.090.080.070.060.050.040.030.0

nosnruterfoytilitaloV

Tota

l loa

n vo

lum

e

noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa

rotceS snaol1

Figure 6.14: Total loan volume as a function of σ1

This figure illustrates the dependence of the equilibrium total loan volumes L on the volatility ofreturns on first-sector loans, σ1. Total loan volumes are shown for five different regimes: thelaissez-faire equilibrium, the Standardized Approach with both equal and different risk weights, andthe VaR approach with confidence levels p = 1.0% and with p = 0.1%. The built-in chart highlightsthe complete σ1-L∗ chart for all σ1 considered.

depositors also enjoy higher success probabilities despite increased portfolio risks

due to the reduced leverage. For the Standardized Approach with different risk

weights, results are mixed.

To sum up, lending under the VaR approach is not pro-cyclical for µ1 ≤ µ2 and

for WH being sufficiently high, and even counter-cyclical for µ1 > µ2. If loans are

equally weighted, i.e. c1 = c2, there is no pro-cyclical effect on loan volumes either.

Crude distinctions of risks by fixed risk weights, in this example by setting c1 = 0.5

and c2 = 1, may accelerate the sensitivity in total lending as well as in sector-wise

lending.

6.6.3 Return Volatility Shocks

Figure 6.14 shows the comparative static results for volatility shocks. The

unregulated total loan volume L∗ strictly decreases in σ1 reflecting the risk-aversion

of the depositors. The total loan volumes under the VaR approach strictly decrease

as well, whereas two effects can be identified. First, as without regulation, the

equilibrium loan-allocation rate strictly decreases in σ1 as loans to Sector 1 become


0001

0051

0002

0052

0003

0053

0004

0054

0005

0055

0006

0056

0007

0057

0008

040.0530.0030.0520.0020.0510.0010.0500.0000.0nosnruterfoytilitaloV

Tota

l loa

n vo

lum

enoitalugero/w

ehttaRaVlevel%1.0

rotceS snaol1

Figure 6.15: Total loan volume as a function of σ1, Table 6.2, Panel A

This figure illustrates the dependence of the equilibrium total loan volume L on the volatilityof returns on first-sector loans, σ1. Results are based on the parametrization given by Table6.2, Panel A. Total loan volumes L are shown for two different regimes: the laissez-faire equilibriumand the VaR approach with confidence level p = 0.1%.

less favorable. As the regulatory constraint function w(`) strictly increases in the

loan-allocation rate ` for ` ∈ [0, `σmin ], we obtain a negative effect in total, i.e. the

total loan volume shrinks. Second, the regulatory constraint function w(`) also

strictly declines in σ1, notably because of ρ > 0, which enhances the former effect.

The drop in the unregulated total loan volume is stronger than those in the total

loan volumes resulting from the VaR approach. As a consequence, regulation is

non pro-cyclical concerning total lending. Figure 6.15, which is produced using the

alternative parametrization from Table 6.2, Panel A, gives a more detailed view on

total loan volumes under return volatility shocks and thus illustrates the dampening

effect of VaR regulation. If the expected economic situation worsens, the risk-averse

depositor becomes reluctant to lend to the bank such that the total loan/deposit

volume shrinks to a point where regulation ceases to bind from a fixed value of σ1

on, in this example from σ1 ≈ 0.202 on.

In the base case scenario we do not explicitly consider the case where the household

is constrained by its initial wealth WH , but in the scenario shown in Figure 6.15 we

do. Wealth constraints may lead to pro-cyclicality as follows:

For σ1 ≤ 0.0083, the bank would like to attract more deposits, even under regulation,


0

004

008

0021

0061

0002

0042

21.011.001.090.080.070.060.050.040.030.0


noitalugero/w

dezidradnats

1=2c=1c

dezidradnats

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

htiwhcaorppa

htiwhcaorppa

Volu

me

of L

oan

portf

olio

2

rotceS snaol1

Figure 6.16: Loan volume to Sector 2 as a function of σ1

This figure illustrates the dependence of the equilibrium loan volumes granted to Sector 2 on thevolatility of returns on first-sector loans, σ1. Loan volumes L2 are shown for five different regimes:the laissez-faire equilibrium, the Standardized Approach with both equal and different risk weights,and the VaR approach with confidence levels p = 1.0% and with p = 0.1%.

than the household is able to provide. Hence, in spite of binding regulation, there

is no effect on the total loan volume and regulation is not pro-cyclical according to

Definition (2.2.2). For σ1 in [0.0083, 0.0106], the unregulated bank is constrained

by the household’s budget whereas the regulated bank is only restricted by the

Regulatory Requirement (6.5.2). Consequently, we obtain L∗ = WB + WH without

regulation, but a risk-sensitive total loan volume under regulation. Hence, the VaR-

based regulation is pro-cyclical. If the bank were regulated by equal risk weights,

we would not observe a pro-cyclical effect either. It is not until [0.0106, 0.0202] that

the dampening effect through regulation begins to hold.

Though the binding regulation for σ1 ≤ 0.0083 is not reflected by total loan volumes,

it is reflected by the different the loan-allocation rates and the different deposit

interest rates, where `V > `S andRVD < RS

D holds. If the household is not constrained

by its initial wealth WH under either regimes, both loan-allocation rates are equal

and approach the variance-minimum loan-allocation rate `σmin with decreasing σ1.

In principle, we should always expect one of the three effects of regulation on total

lending as described above if σ1 varies for the following reasons:

First, as the depositor’s budget is bounded (as short selling is excluded), the level


00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

7711.05501.03390.01180.00960.08650.06440.05230.0


0000.0

5000.0

0100.0

5100.0

0200.0

5200.0

Dep

osit

retu

rn v

olat

ilitie

s un

der V

aR



ehttaRaVlevel%0.1

)snaol(

ehttaRaVlevel%1.0

)snaol(

ehttaRaVlevel%0.1

)stisoped(

ehttaRaVlevel%1.0

)stisoped(

Ret

urn

vola

tiliti

es

rotceS snaol1

Figure 6.17: Return volatilities as a function of σ1

This figure illustrates the dependence of the equilibrium return volatilities on the volatility of returnson first-sector loans, σ1. Solid and dashed lines represent volatilities of returns on deposits, σD.Diamonds and crosses represent volatilities on total loans, σ. Return volatilities are shown forthree different regimes: the laissez-faire equilibrium and the VaR approach with confidence levelsp = 1.0% and p = 0.1%.

of shocked expectations can always be such that deposit supply is restricted by the

initial wealth independent of whether the bank is regulated or not.9

Second, the regulated equilibrium deposit volume DV will be already below WH

for lower values of σ1 than the unregulated volume D∗ since DV ≤ D∗ holds by

definition. By the assumption of strictly decreasing equilibrium deposit volumes, this

property creates a pro-cyclical effect of the VaR regulation according to Definition

(2.2.1).

Third, an expectation shock can take values such that the household’s initial

wealth WH does not constrain equilibrium total loan/deposit volumes anymore.

If furthermore curvature does not change too much10, the curve representing the

unregulated total loan/deposit volume is steeper than that under regulation and

9For this reasoning, the assumption that the unconstrained equilibrium deposit volume strictlydecreases in σi, i = 1, 2 is crucial. The strictly monotonic decline must hold for both regulatedand unregulated volumes. Note that Result 31 does not account for the endogenous effects on ànd RD in equilibrium , however.

10For the importance of curvature on cycle-related effects, we refer to the example shown inFigure 6.10 concerning equity shocks.


00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0

21.011.001.090.080.070.060.050.040.030.0




1=2c=1c)stisoped(





)snaol(1=2c=1c


1=2c,5.0=1c)snaol(

Ret

urn

vola

tiliti

es

rotceS snaol1

Figure 6.18: Return volatilities as a function of σ1

This figure illustrates the dependence of the equilibrium return volatilities on the volatility of returnson first-sector loans, σ1. Solid and dashed lines represent volatilities of returns on deposits, σD.Diamonds and crosses represent volatilities on total loans, σ. Return volatilities are shown forthree different regimes: the laissez-faire equilibrium and the Standardized Approach with both equaland different risk weights.

thus there is no longer any pro-cyclical impact on lending.11

Apart from the total loan volumes under the VaR approach, Figure 6.14 depicts the

total loan volumes under the Standardized Approach. In case of c1 < c2 , the total

loan volume decreases in σ1 as well and is not pro-cyclical either. For σ1 < 0.069,

the total loan volume does not react to changes in σ1 and the loan-allocation rate

`S remains close to 100%. Consequently, the total loan volume is not pro-cyclically

affected. If σ1 > 0.069, the regulated bank reduces its loan-allocation rate and total

lending shrinks under regulation. Then it is conceivable that it can be pro-cyclical,

though it is not in this example.

The stronger decline in the loan-allocation rate `S compared to the decrease in the

unregulated loan-allocation rate, `∗, results in a stronger deflation of LS1 compared

to L∗1. Thus, on the level of sectors, we observe a pro-cyclical effect. This sharper

decline in `S, in turn causes the loan volume granted to firms from the second sector

11As the analysis in Section 4.3.1 suggests, these arguments also hold for shocks in correlations.Within the normal framework, we refer to the next section, Section 6.6.4. The preceding argumentshold for shocks in expected returns as well (recall Figures 6.11 and 6.12 for the normal frameworkand Figures 4.19 and 4.20 concerning the Bernoulli framework).


000,1

000,3

000,5

000,7

000,9

000,11

000,31

000,51

000,71

000,91

000,12

%09.6%07.6%05.6%03.6%01.6%09.5%07.5%05.5%03.5

etartseretnitisopeD

Dep

osit

volu

me

64230.0

25630.0

75040.0

36440.0

96840.0

57250.008650.0

79860.0

02580.0

7121.0

muirbiliuqellafosucolsemulovtisoped

Vola

tility

of r

etur

ns o

n S

ecto

r 1 lo

ans

Figure 6.19: Risk effects in equilibrium on deposits

This figure illustrates the influence of the volatility of returns on first-sector loans, σ1, on theequilibrium deposit volume given optimally chosen loan-allocation rates. Deposit volumes are shownas functions of the deposit interest rate for different levels of σ1. The curve on the top depictsDs(`∗, RD) for σ1 ≈ 0.03246 whereas the curve on the bottom represents Ds(`∗, RD) for σ1 ≈0.1217. The grey curve is the locus of all equilibrium deposit volumes and deposit interest rates.This figure exclusively refers to the laissez-faire equilibrium.

to strictly increase. In contrast, the unregulated bank reduces also lending to firms

belonging to Sector 2, reflecting its prudence due to the risk-averse depositor and

its deposit-supply function to the rising loan return volatility σ1. Thus, there is a

counter-cyclical effect concerning lending to Sector 2. The second phenomenon is

illustrated by Figure 6.16.

Deposit interest rates strictly increase in σ1 under all regimes except for the

Standardized Approach with both risk weights being equal to one (not shown). The

equilibrium loan-allocation rates under these four regimes decrease in σ1. Yet, the

loan portfolio’s return volatilities strictly increase (cf. Fig. 6.17 and 6.18), implying

a strict partial decrease in deposit supply due to Result 31. Though lower loan-

allocation rates result in higher expected portfolio returns µ and though deposit

interest rates are raised, the total effect of all these changes on the deposit volume,

and hence the total loan volume (cf. Fig. 6.15) is negative. Figure 6.19 illustrates this

issue in the RD −D plane as the loan-allocation rate ` is fixed to `∗, i.e. to the rate

that is optimally chosen by the unregulated bank. Thus, the deposit-supply function

Ds(`∗, RD) exclusively depends on the deposit interest rate for fixed values of σ1.


0

004

008

0021

0061

0002

0042

0082

0023

0063

0004

0044

%03%82%62%42%22%02%81%61%41%21%01%8%6%4%2%0

snruterrotceshtobneewtebnoitalerroC

Tota

l loa

n vo

lum

enoitalugero/w

dezidradnatsch withaorppa1=2c=1c

dezidradnatsh withcaorppa

1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

Figure 6.20: Total loan volume as a function of the inter-sectoral correlation

This figure illustrates the dependence of the equilibrium total loan volumes L on the inter-sectoralreturn correlation ρ. Total loan volumes L are shown for five different regimes: the laissez-faireequilibrium, the Standardized Approach with both equal and different risk weights, and the VaRapproach with confidence levels p = 1.0% and p = 0.1%.

Ds(`∗, RD) is further depicted for different values of σ1 to highlight the reduction in

deposit volume by increasing σ1.

Except for the Standardized Approach with both risk weights being equal, regulation

raises the probability of full deposit redemption. This holds true even for the

Standardized Approach with different risk weights where the volatility of the loan

portfolio is greater than it is the case without regulation (cf. Fig. 6.18). But

even under the regime with c1 = c2 = 1, this probability exceeds its unregulated

counterpart from σ1 ≈ 0.811 on.

6.6.4 Correlation Shocks

Correlation shocks affect the loan return volatility σ similarly as volatility shocks

with respect to single sectors. Consequently, the effects on magnitudes that are

dependent on both loan portfolios’ returns are similar. In contrast to the volatility

shocks considered, compensating the shock by shifting portfolio weights as seen

in the last section (cf. Fig. 6.16) does not happen if the correlation remains at

intermediate levels. Under the scenario considered in Figure 6.21, loan-allocation


rates move strongly only when correlations exceed ±0.7. However, from an

empirical point of view, these values are unrealistic as high for correlations on

loan redemptions. Having the Merton-framework in mind, these values are also

problematic from a theoretical point of view. As Gersbach/Lipponer (2003, p. 364f)

show, default correlations are always positive and are bound by 2π· arcsin(ρreturn),

where ρreturn is the firms’ asset correlation, if both firms are equal. Given their

assumptions, a default correlation may only be above 0.7 if the asset-return

correlation of two equal firms exceeds 0.891. Likewise, a default correlation of 0.31

would be associated with an asset correlation of 0.454. Hence, it seems reasonable

to consider correlations to be from the interval [0, 0.31] for the comparative static

analysis in the base case.

As with increasing σi, an increase in ρ result in a considerably higher σ, and in

a partial decrease both in the unregulated total loan/deposit volume and in the

VaR-constrained total loan/deposit volume because of Result 31 and because of the

Regulatory Restriction (6.5.2), respectively. Figure 6.20 shows that these effects

carry over to the equilibrium as well. As in the case of increasing volatilities, the

unregulated total loan volume is more affected by increasing correlations than the

VaR-regulated volume. Hence, VaR-based regulation does not enhance cyclicality

under correlation shocks either.

The total loan volume does not change under the Standardized Approach, as the

different risk weights do not account for symmetrical shifts in risks and the loan-

allocation rate is not affected in noteworthy magnitudes. Hence, this kind of

regulation does not affect lending in a pro-cyclical way.

Concerning pro-cyclicality, we observe patterns that are analogous to those in the

case of volatility and expected return shocks. Figures 6.20 and 6.21 suggest that

the regulated and the unregulated total loan volumes come closer to each other

as the return correlation becomes higher while a decreasing ρ leads to a stronger

rise in the unregulated total loan volume than in the regulated total loan volumes.

Furthermore, Figure 6.21 illustrates that regulation can be pro-cyclical with respect

to the total loan volume if the depositor is wealth-constrained. As with shocks to σ1,

the regulated equilibrium deposit volume DV will be already below WH for lower

values of ρ than the unregulated volume D∗ since DV ≤ D∗ holds by definition.

By the strictly decreasing equilibrium deposit volumes, this property creates a pro-

cyclical effect of the VaR regulation.

Figure 6.22 shows that the volumes granted to firms from the first sector are

reduced if the correlation between the firms of different sectors decreases. This


0001

0051

0002

0052

0003

0053

0004

0054

0005

0055

0006

0056

0007

0057

0008

8.07.06.05.04.03.02.01.00.01.02.03.04.05.06.07.08.09.0snruterrotceshtobneewtebnoitalerroC

Tota

l loa

n vo

lum

enoitalugero/w

ehttaRaVlevel%1.0

Figure 6.21: Total loan volume as a function of ρ, Table 6.2, Panel A

This figure illustrates the dependence of the equilibrium total loan volume L on the inter-sectoralreturn correlation ρ. Results are based on the parametrization given by Table 6.2, Panel A. Totalloan volumes L are shown for two different regimes: the laissez-faire equilibrium and the VaRapproach with confidence level p = 0.1%.

behavior results from decreasing total loan volumes (cf. Fig. 6.20) while the loan-

allocation rates remain almost constant (not shown) within the range of values of

return correlations considered. More specifically, 59.60% < `∗ < 59.80% holds,

and `V ranges under each of both VaR constraint between 59.68% and 59.79%. As

Figure 6.22 indicates, loan-allocation rates strongly differ from their unregulated

counterparts under the Standardized Approach. The magnitudes of the differences

depends on whether the fixed risk weights distinguish the two types of loans: if

c1 = c2 = 1, the banks prefers granting loans to the riskier sector implying

0 ≤ `S < 2.26%, while `S takes values from 88.59% to 89.70% if c1 = 0.5 and

c2 = 1 holds. The associated risks are indicated by the return volatilities concerning

total loans and deposits, respectively, as shown by Figures 6.24 and 6.25, which

are discussed below. If the alternative parametrization from Table 6.2, Panel A,

is considered, however, the unregulated loan-allocation rate `∗ strongly increases,

namely from 62.8% to 65.1% for 0.0% ≤ ρ ≤ 30%. Furthermore, the total loan

volume is less sensitive to changes in ρ for 0.0% ≤ ρ ≤ 30% than it is the case for L∗

under the base case. In numbers, we have ∆L∗

∆ρ= −1060.3 under the parameter values

from Table 6.2, Panel A, while ∆L∗

∆ρ= −3492.85 holds in the base case as given by

Table 6.1. Consequently, L∗1 is also less sensitive to shocks in ρ for 0.0% ≤ ρ ≤ 30%


0

005

0001

0051

0002

0052

0003

%92%62%32%02%71%41%11%8%5%2%0snruterrotceshtobneewtebnoitalerroC

0

001

002

003

004

005

00oitalugero/w n


1=2c=1c


1=2c,5.0=1c

ehttaRaVlevel%0.1

ehttaRaVlevel%1.0

Volu

me

of L

oan

portf

olio

1

6

Figure 6.22: Loan volume to Sector 1 as a function of ρ

This figure illustrates the dependence of the equilibrium loan volumes granted to Sector 1 on theinter-sectoral return correlation ρ. Loan volumes L1 are shown for five different regimes: thelaissez-faire equilibrium, the Standardized Approach with both equal and different risk weights, andthe VaR approach with confidence levels p = 1.0% and with p = 0.1%.Left-hand scale: w/o regulation and the Standardized Approach with different risk weights.Right-hand scale: both VaR approaches.

in the scenario given by Table 6.2 than in the base case.

Furthermore, if the household is wealth-constrained, pro- and counter-cyclical effects

in lending concerning a single portfolio Li, i = 1, 2, may arise, as shown in Figure

6.23. For−90% ≤ ρ < −77%, LV1 and L∗1 strictly increase where LV1 is steeper. Thus,

the loan volume lend to firms from the first sector becomes pro-cyclical through

regulation. For −77% < ρ < −62%, the regulated loan volume LV1 strictly decreases

while L∗1 further strictly increases. Thus, regulation has a counter-cyclical effect

according to Definition (2.2.3). From ρ = −63% on, the loan volumes lend to firms

from the first sector strictly decrease under both regimes, whereas the slope of the

unregulated volume, L∗1 with respect to changes in ρ is larger. Thus, regulation is

not pro-cyclical from ρ = −63% on.

On the domain where regulation has a counter-cyclical effect, i.e. for ρ ∈[−77%,−63%], the household is restricted by its initial wealth WH in the laissez-faire

economy, while it is not if the bank is regulated. As a consequence, the regulated

total loan volume is already downward-sloping for increasing ρ. The unregulated

total loan volume, however, is fixed to L∗ = WH +WB. As the loan-allocation rate


0001

0051

0002

0052

0003

0053

0004

%07%05%03%01%01-%03-%05-%07-%09

n between both sector returnsoitalerroC

Volu

me

of L

oan

portf

olio

1o/w noitaluger

ehttaRaVlevel%1.0

-

Figure 6.23: Loan volume to Sector 1 as a function of ρ, Table 6.2, Panel A

This figure illustrates the dependence of the equilibrium loans volumes granted to Sector 1 onthe inter-sectoral return correlation ρ. Results are based on the parametrization given by Table6.2, Panel A. Loan volumes L2 are shown for two different regimes: the laissez-faire equilibriumand the VaR approach with confidence level p = 0.1%.

`∗ strictly increases in ρ without regulation,12 the loan volume L∗1 strictly increases

as well for ρ ∈ [−77%,−63%].

Concerning risk-taking, we refer to Figures 6.24 and 6.25 which show the return

volatilities of the loan portfolio and of the deposit for intermediate values of

correlations in the base case. As in the other shock scenarios, the loan-allocation

rates are not significantly altered by the VaR regulation. Thus, loan return

volatilities are very close to their unregulated counterparts. Since the VaR regulation

results in lower deposit volumes and deposit interest rates, deposit return volatilities

are lower under regulation, too. Only under the Standardized Approach adverse

effects occur. Since binding regulation always reduces the bank’s leverage, the buffer

effect of bank capital is also strengthened under the Standardized Approach such

that deposit return volatilities scatter less than the associated loan return volatilities.

Furthermore, the probabilities of full deposit redemption are also higher under both

forms of regulation compared to the unregulated regime, whereas those under the

VaR approach are the highest.

Analogous to volatility shocks, the increasing return volatilities σ and σD,

12Also `V strictly increases with decreasing ρ.


00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

%92%62%32%02%71%41%11%8%5%2%0

Ret

urn

vola

tiliti

es

0000.0

5000.0

0100.0

5100.0

0200.0

5200.0

Dep

osit

retu

rn v

olat

ilitie

sun

der V

aR


ehttaRaVlevel%0.1

)stisoped(

ehttaRaVlevel%1.

stisoped

n between both sector returnsoitalerroC


ehttaRaVlevel%0.1

)snaol(

ehttaRaVlevel%1.0

)snaol(

0)(

Figure 6.24: Return volatilities as a function of ρ

This figure illustrates the dependence of the equilibrium return volatilities on the inter-sectoralloan-return correlation ρ. Solid and dashed line represent volatilities of returns on deposits, σD.Diamonds and crosses represent volatilities on total loans, σ. Return volatilities are shown forthree different regimes: the laissez-faire equilibrium and the VaR approach with confidence levelsp = 1.0% and p = 0.1%.

respectively, lead in both scenarios considered in this analysis to higher deposit

interest rates which has also been confirmed under the framework with the Bernoulli

distribution (cf. Figure 4.13).

6.7 Summary

This numerical analysis has shown that, by and large, the results obtained in Chapter

4 carry over to this framework which considers normally distributed aggregate

loan redemptions. The effects regulation can have depend on the kind of capital

requirements and the exogenous shock as well as the endogenous variable considered.

Under past shocks that impaired the bank’s equity once, a binding regulation

affects total lending pro-cyclically. This again holds true for both the Standardized

Approach and the VaR approach. Assigning different risk weights may partially

lead to higher volumes granted to a single sector under regulation than without

regulation, thus questioning a general pro-cyclical impact on lending on the

disaggregate level (cf. Fig. 6.8). A similar effect has been observed under the

6.7. SUMMARY 219

00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

01.0

11.0

21.0

%03%82%62%42%22%02%81%61%41%21%01%8%6%4%2%0

snruterrotceshtobneewtebnoitalerroC

Ret

urn

vola

tiliti

es



1=2c=1c)stisoped(





1=2c=1c


1=2c,5.0=1c)snaol(

)snaol(

Figure 6.25: Return volatilities as a function of ρ

This figure illustrates the dependence of the equilibrium return volatilities on the inter-sectoralloan-return correlation ρ. Solid and dashed lines represent volatilities of returns on deposits, σD.Diamonds and crosses represent volatilities on total loans, σ. Return volatilities are shown forthree different regimes: the laissez-faire equilibrium and the Standardized Approach with both equaland different risk weights.

Bernoulli framework (cf. Fig. 4.5).

Furthermore, we provided an example emphasizing that the sensitivity of the

unregulated total loan volume is crucial for sustaining pro-cyclical effects. The

more concave the total loan/deposit volume is with respect to the bank’s initial

equity, the more likely it is that there is a domain on which regulation dampens the

sensitivity of the total loan volume. The notion of pro-cyclicality on average thus

becomes a suitable term.

If the agents’ expectations are shifted, total lending mostly reduces sensitivities

under binding regulation. But there are exceptions. Concerning shocks in the

volatility σ1, the sector-wide loan volume LS1 granted under the Standardized

Approach with different risk weights deflates stronger than its unregulated

counterpart. On the other side, we observe a counter-cyclical reaction in the loan

volume granted to Sector 2, L2: the regulated volume strictly increases on this

domain, while the unregulated volume strictly decreases.

Furthermore, wealth-restrictions with the household may result in both pro- and

counter-cyclical effects.


The VaR approach always reduces risks in terms of deposit return volatilities and

the bank’s probability of bankruptcy which is in line with Dangl/Lehar (2004)

though their setting is different from ours, in particular in terms of the bank’s

binary portfolio choices that can be revealed only by auditing from time to time.

The adverse effects that were present under the VaR approach within the Bernoulli

framework do not recur here as loan redemptions are no longer lumpy. Consequently,

the regulatory constraint becomes smooth and extreme allocations in favor of either

sector can no longer be observed. Concerning risk, the Standardized Approach offers

mixed effects as in the Bernoulli-distribution based framework.

So far, the question of how a shock that has once occurred may further affect the

economy could not be addressed. In particular, we wish to examine if expectation

shocks have lasting effects. Consequently, we will extend this model to two periods

in the following chapter.

Chapter 7

The Two-Period Model

7.1 Timeline and Decisions

In this chapter, we extend the model from Chapter 6 to two periods. Decisions are

made at the beginning of each period, at t = 0 and at t = 1. All contracts last

for one period each. Thus the household decides both at t = 0 and t = 1 on its

portfolio composition and the bank does the same. We assume that the institutional

framework from the one-period model holds for each period separately.

Both the bank and the household are perfectly informed about each other’s sets of

feasible decisions, as in the one-period models. They also have rational expectations

regarding their own and the other’s reaction in t = 1 to decisions made at t = 0. As

a consequence, the household and the bank will base their decisions at t = 0 on the

possible equilibria in t = 1. Likewise, the household and the bank will base their

decisions at t = 1 on what happened then.

Therefore, the model will, as usual, be solved backwards. First, the decisions at t = 1

given any decisions and outcomes from the first period will be derived, and then the

decisions and the equilibrium in t = 0, given all possible equilibria in t = 1, will

be determined. Given the institutional framework, there are thus no incentives to

deviate from the decisions thus derived and the solution is subgame-perfect. As the

stages of this game are interpreted as a timeline, the equilibria can also be considered

to be time-consistent: There is no point in time at which either the household or

the bank can credibly threaten to deviate (cf. Mas-Colell/Winston/Green, 1995,

p. 267ff).

We begin by analyzing the model in the second period. Expectations formed at

t = 1 are indexed by 1, those formed at t = 0 by 0. The model’s parameters

221

222 CHAPTER 7. THE TWO-PERIOD MODEL

referring to the returns on the bank’s loan portfolio are indexed either by 1 or 2,

depending on the period they refer to, or, equivalently, the point of time at which

they are realized. Decision variables such as the loan-allocation rates, the deposit

interest rates, and the deposit volumes are indexed by the point of time at which

the decision was made; that is, either by t = 0 or t = 1. The time indices of the

riskless rates refer, in analogy to the deposit interest rates, to the point in time

at which the household starts to save its remaining funds, and not to the point

in time at which the riskless rate is realized. Thus, Rf,t denotes the riskless rate

starting at t and ending at t + 1. The same applies to the deposit interest rates

RD,t, whereas the loan-portfolio return xt indicates that it was realized in t while

the associated portfolio composition was chosen at t− 1. Likewise, the symbols µt,

σt, and ρt denote the expected portfolio return, the portfolio’s return volatility and

the correlation of the sector returns associated with the returns to be realized in t.

In particular, µ1, σ1, and ρ1 denote the initial distributional parameters referring

to the first period. The parameters characterizing the return distributions of the

respective subportfolios are indexed by i, t, i = 1, 2 and t = 1, 2. Hence, µ1,1 is the

expected return on the loans granted to Sector 1, as expected at t = 0 for the first

period. We assume that portfolio returns are independent across both periods. At

the final date t = 2, the realizations of the portfolio returns x2 result in the bank’s

final wealth WB,2 and in the household’s final wealth WH,2.

Figure 7.1 outlines the wealth timeline. Each state is characterized by the bank’s

amount of equity and the household’s wealth at each point in time t. Though there

is a continuum of states at each point in time t as gross returns on the bank’s

loan portfolio are normally distributed, these states can be summarized by a few

nodes that reflect the main event sets. These main event sets are characterized by

whether or not the bank can redeem the deposits as promised. Failure comprises

both the set of states in which the household receives positive payments that are

based on positive residual claims on the bank’s assets and the set of states in which

the household receives zero payments. If the bank fails after the first period, the

household can only invest its funds in the riskless asset for the second period.

Consequently, there are only two critical nodes in t = 2 following both nodes in

t = 1 that symbolize the failure events. The paths linking the nodes are tagged by

the conditional probabilities that can be assigned to each respective node, i.e. set

of states, to which the path points.

Analogous to Definition (6.2.3), bdz ,2 denotes the bank’s solvency barrier at the end

of the second period, and bdz ,1 denotes the solvency barrier after the first period.

7.1. TIMELINE AND DECISIONS 223

Figure 7.1: Wealth timeline

(WB,0

WH,0

) u%%%%%%%%%%%%%%%%%%

eeeeeeeeeeeeeeeeee

u

u

u

1

1

@@@@@@@@@@@@

u

u

u

u

u

p1,1

p2,1

p3,1

p1,2

p2,2

p3,2

t = 0 t = 1 t = 2

(WB,1 = x1(D0 +WB,0)−D0RD,0

WH,1 = D0RD,0 + (WH,0 −D0)Rf,0

)

(WB,1 = 0WH,1 = x1(D0 +WB,0) + (WH,0 −D0)Rf,0

)

(WB,1 = 0WH,1 = (WH,0 −D0)Rf,0

)

(WB,2 = 0WH,2 = (WH,0 −D0)Rf,0Rf,1

)

(WB,2 = 0WH,2 = WH,1Rf,1

)

(WB,2 = 0WH,2 = (WH,1 −D1)Rf,1

)

(WB,2 = 0WH,2 = x2(D1 + WB,1) + (WH,1 −D1)Rf,1

)

(WB,2 = x2(D1 + WB,1)−D1RD,1 ≥ 0WH,2 = D1RD,1 + (WH,1 −D1)Rf,1

)

withp1,t = Φ(bdz,t)p2,t = Φ(µt

σt)− Φ(bdz,t)

p3,t = 1− Φ(µt

σt)

t = 1, 2


Formally, these standardized barriers can be expressed as

bdz ,1 =µ1 − D0RD,0

D0+WB,0

σ1

,

bdz ,2 =µ2 − D1RD,1

D1+WB,1

σ2

. (7.1.1)

The solvency barrier at the end of the second period is a random variable given the

available information at t = 0 and thus tagged by a tilde. In particular, all main

components of bdz ,2 are random variables as they depend on the realizations of the

state variables WB,1 and WH,1. At t = 1 uncertainty concerning bdz ,2 is resolved and

we can simply write bdz ,2. Likewise, any other realizations of random variables will

be written without a tilde as well.

As the household is faced in t = 1 with the same portfolio decision as in the

one-period model, the expected utility from (6.3.1) can be re-formulated as the

household’s objective function at t = 1. Hence we obtain

E1

[uH(WH,2) |(WB,1,WH,1)

]= −e−γ(WH,1−D1)Rf,1 · Φ(−µ2

σ2

)

−e−γ[µ2(D1+WB,1)+(WH,1−D1)Rf,1−γσ22(D1+WB,1)2] ·

·

Φ(

µ2 − γσ22(D1 +WB,1)

σ2

)− Φ(µ2 − D1RD,1

D1+WB,1− γσ2

2(D1 +WB,1)

σ2

)

−e−γ[D1RD,1+(WH,1−D1)Rf,1] · Φ(bdz ,2) ,

(7.1.2)

as expectations are formed conditional on realizations of WB,1 and WH,1 at t = 1.

As a consequence, the household’s deposit supply in t = 1 has properties that are

analogous to those in the one-period model, except for WB,1 = 0, since the bank is

considered bankrupt in this state and is no longer allowed to operate. We assume

RD,1|x1=D0RD,0D0+WB,0

:= 0 (7.1.3)

resulting in

Ds1(`1, RD,1)|

x1=D0RD,0D0+WB,0

= 0

due to Result 32. This assumption prevents the household from depositing funds

at the bank after the bank has already defaulted. In particular, Condition (7.1.3)


ensures that

E1

[uH(WH,2) |WB,1

]∣∣∣x1=bdx,1

= − e−γWH,1Rf,1

holds, as derived in Appendix D.2 in connection with the proof of Result 36.

Condition (7.1.3) leads to a jump in the second period’s deposit-supply function

at WB,1 = 0. The supply function is continuous from below (and equal to zero) in

terms of realized equity WB,1 or, equivalently, in terms of portfolio realizations x1.

A bankruptcy of the bank in t = 1 results in the following expected losses for

both the bank and the household: First, the bank can no longer generate expected

wealth from intermediation in the second period as it will be closed. Second, after

the bank’s closure, the household can only invest its funds in the risk-free asset.

Hence, the household also incurs a loss in expected wealth in the second period if

the bank goes bankrupt, as it must forgo the opportunity to invest funds in risky

deposits.

Analogous to the one-period model, (6.2.1), at t = 1 the bank’s expectation can be

expressed as follows, conditional on WB,1 and WH,1,

E1

[WB,2(`1, RD,1) |(WB,1,WH,1)

]= σ2 · (D1 +WB,1) · ϕ(bdz ,2)

+ [µ2 · (D1 +WB,1)−D1 ·RD,1] · Φ(bdz ,2)

as its final wealth WH,2. Thus, the bank’s conditional decision in t = 1 can be

characterized as discussed in the one-period model if WB,1 > 0 holds.

The equilibrium choice at t = 1 will be denoted by (`∗1, R∗D,1) if the bank is

unregulated. The superscript V indicates equilibrium choices under VaR-based

regulation. Solving the model backwards, optimal choices made at the beginning

of the second period are contingent on a continuum of potential decisions in t = 0

and on the realized portfolio return x1. All these parameters affect (`∗1, R∗D,1) via

the bank’s equity WB,1 and via the households’s wealth WH,1. If the household is

not constrained in t = 1, bank capital WB,1 remains as the only link between the

decisions made in the first and the second periods. The loan-allocation rate chosen

at t = 0, in turn, indirectly influences WB,1 in two ways: first via D∗0 and second

via the distribution of the portfolio returns x1. As variables in t = 1 depend on

WB,1, which depends inter alia on the realized return x1, all variables in t = 1

are random variables in t = 0. Thus, the household and the bank managers will

base their expectations on the second period’s equilibria when taking decisions at

t = 0. Formally, the connection between the second-period decision variables, here


the portfolio-allocation rate, and the first-period variables is as follows:

˜∗1 = `∗1(WB,1) = `1(x1, D0, RD,0)

= `1 (x1, Ds0(`0, RD,0), RD,0)

= `1 (x1, D0(`0(µ1,1, µ2,1, σ1,1, σ2,1, ρ1, ·), RD,0(·)), RD,0(·)) ,

(7.1.4)

where RD,0(·) potentially depends on the same parameters as `0(·).

Given the optimal decisions in t = 1,(`∗1, R

∗D,1

), the bank owners’s expectations at

t = 0 over final wealth in t = 2 are

E0

[E1

[WB,2(˜∗

1, R∗D,1)

∣∣∣(WB,1, WH,1)]]

=

∞∫

D0RD,0D0+WB,0

E1

[WB,2(`∗1, R

∗D,1) |(WB,1,WH,1)

]· 1

σ1

· ϕ(x1 − µ1

σ1

) dx1 .

A further analytical evaluation of the integral is not possible. Even if both sector-

wise expected returns µ1,i are equal and the optimal loan-allocation rate `∗1 becomes

a constant, the interest rate on deposits will still depend on the bank’s capital in

t = 1 in a manner that makes it analytically intractable. Hence, characterizations

of the decision in t = 0 can only be expressed numerically.

Assume that the household is not constrained by its wealth WH,1 if it can decide

to invest in bank deposits at t = 1; that is, Du1 < Ds

0 · RD,0 + (WH,0 − Ds0) · Rf,0

holds. As a consequence, the expectations formed at t = 1 will be only conditional

on WB,1. Then the household’s expected utility in t = 0 over final wealth WH,2 in

t = 2 is given by


E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]=

=

∞∫

D0RD,0D0+WB,0

−e−γ[D0RD,0+(WH,0−D0)Rf,0−D1]Rf,1 ·

1− Φ(

µ2

σ2

)

+ e−γ[µ2(D1+WB,1)− 12γσ2

2(D1+WB,1)2] ·

·

Φ(

µ1 − γσ21(D1 +WB,1)

σ1

)− Φ(µ1 − D1RD,1

D1+WB,1− γσ2

1(D1 +WB,1)

σ1

)

+ e−γD1RD,1 · Φ(bdz ,2)

·

1

σ1

· ϕ(x1 − µ1

σ1

) dx1

+

D0RD,0D0+WB,0∫

0

−e−γ[x1(D0+WB,0)+(WH,0−D0)Rf,0]Rf,1 · 1

σ1

· ϕ(x1 − µ1

σ1

) dx1

+

0∫

−∞

−e−γ(WH,0−D0)Rf,0Rf,1 · 1

σ1

· ϕ(x1 − µ1

σ1

) dx1 .

The first integral represents all states WB,1 in t = 1 where bank debt is fully

repaid to the household. The second integral describes all states in which positive

aggregate loan redemptions are received by the bank but are too low to fully repay

the obligations to the household. The third integral summarizes all states in which

all borrowers fully default on their loans and the household is only paid off from the

riskless asset.

As with the representation of the bank’s expected equity value, the first integral

cannot be simplified further if one solves for equilibrium. The reason is that the first

integral represents the part of expected utility that has been formed conditional on

WB,1. In particular, D1 is a random variable dependent on WB,1. The same applies

to RD,1 and ˜1, respectively. The latter variable affects µ2 and σ2 such that µ2 in

fact represents a conditional expectation and σ2 a conditional volatility.

As this dependence is analytically unknown, a further analytical determination is

not possible. As the bank will have defaulted in the other two sets of states, no

random variables referring to the second period need be considered and both of the

latter integrals can be further simplified. Expectations in t = 0 over final wealth in

t = 2 can thus be re-formulated as


E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]=

∞∫

D0RD,0D0+WB,0

E1

[uH(WH,2) |WB,1

]· 1

σ1

· ϕ(x1 − µ1

σ1

) dx1

− e−γ[µ1(D0+WB,0)+(WH,0−D0)Rf,0− 12γσ2

1(D0+WB,0)2Rf,1]Rf,1 ·

·

Φ(

µ1 − γσ21(D0 +WB,0)Rf,1

σ1

)− Φ(µ1 − D0RD,0

D0+WB,0− γσ2

1(D0 +WB,0)Rf,1

σ1

)

− e−γ(WH,0−D0)Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

].

(7.1.5)

The integral in the first line will occasionally be denoted by g(D0) while h(D0) will

occasionally refer to the remaining parts.

The conditional equilibrium in t = 1 features the same properties as the equilibrium

in the one-period model examined in the preceding chapter, Chapter 6, if WB,1 is

changed parametrically. Consider the household’s maximization problem in t = 0

given the equilibrium in t = 1:

maxD0

E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]

s.t. 0 ≤ D0 ≤ WH,0 (7.1.6)

(`1, RD,1) =(`∗1(·), R∗D,1(·)

).

First of all, there is always at least one solution to Problem (7.1.6). This result is

shown in Appendix D.1. In fact, the proof is obtained by rephrasing the proof of

the existence of a solution stated in Result 30. To characterize sufficient conditions

for which the household’s deposit-supply function in t = 0 is unique, we define

∆>1 =

(1,

∂R∗D,1∂WB,1

,∂`∗1

∂WB,1

),

∆>E1[uH(·)|·] =

(∂E1 [uH(·)|·]∂WB,1

,∂E1 [uH(·)|·]

∂RD,1

,∂E1 [uH(·)|·]

∂`1

)

if these derivatives exist. In particular, we use ∆1|x1=D0RD,0D0+WB,0

and

∆E1[uH(·)|·]∣∣x1=

D0RD,0D0+WB,0

to denote the limit values for which x1 =D0RD,0D0+WB,0

is

approached from above.1 The expression ∆E1[uH(·)|·] always refers to the conditional

1Note that these terms would vanish if x1 = D0RD,0D0+WB,0

were approached from below because of


(laissez-faire) equilibrium(`∗1(·), R∗D,1(·)

)in t = 1. The definitions of ∆1 and

∆E1[uH(·)|·] are only meaningful if RD,0 > RD,0(`) holds where RD,0(`) is given by

Definition (6.3.7). As in the one-period model `t ∈ [0, 1], t ∈ 0, 1, is assumed.

Result 36. Assume that the conditions

0 > − 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) + h′′(D0)

− γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

+1

σ1

· bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2· e−γWH,1Rf,1 · ϕ(bdz ,1)

+(RD,0WB,0)2

(D0 +WB,0)3· ∆>1 ·∆E1[uH(·)|·]

∣∣x1=

D0RD,0D0+WB,0

· 1

σ1

· ϕ(bdz ,1)

andd[∆>1 ·∆E1[uH(·)|·]

]

dWB,1

< 0

hold, and the respective derivatives exist. In this case Ds0(`0, RD,0) is unique.

The derivation of Result 36 can be found in Appendix D.2.

The second condition states that the sensitivities with respect to the bank’s initial

equity WB,1 become lower as equity increases. These declining sensitivities reflect

the fact that the bank’s equity WB loses its buffer function for the household at

the margin as WB increases. This notion is also reflected by the numerical results

discussed in Section 6.6 concerning equity shocks, i.e. changes in WB. The numerical

examples result in strictly concave curvatures of the total loan volume with respect

to WB for the whole range of values chosen for WB within each of the parameter

variations as given in Table 6.1 and Table 6.2, respectively (cf. Fig. 6.3, 6.9, and

6.10).

As the second condition is a derivative of first-order derivatives, it technically

imposes regularity conditions on the curvature of the household’s expected utility

with respect to its deposit supply, deposits interest rates, portfolio-allocation rates,

and bank equity in t = 1. Details are given in Appendix D.2, cf. especially Formulæ

(D.2.10) to (D.2.15) and (D.2.16).

Assumption (7.1.3).


200

400

600

800

1000

0

1000

2000

3000

4000

5000

6000

1.095 1.105 1.115 1.125 1.135 1.145 1.155 1.165

Tota

l loa

n vo

lum

e (w

/o re

gula

tion)

Expected return on Sector 1 loans in the first period

w/o regulation:first period of thetwo-period model

w/o regulation:one-period model

VaR at the0.1% level:first period of thetwo-period model

VaR at the0.1% level:one-period model

Tota

l loa

n vo

lum

e (V

aR)

Figure 7.2: Total loan volume L0 (L) as a function of µ1,1 (µ1)

This figure illustrates the dependence of the equilibrium total loan volumes on the expected grossreturn of first-sector loans, µ1,1 (µ1). Solid lines represent total loan volumes granted in the firstperiod of the two-period model, while dashed lines indicate total volumes in the one-period model.Total loan volumes are shown for two different regimes: the laissez-faire equilibrium and the VaRapproach with confidence level p = 0.1%.

7.2 Numerical Analysis of Regulatory Impacts

The aim of this section is to determine whether expectation shocks have a lasting

effect on lending and if they accelerate the cyclical nature of lending as soon as a

VaR constraint comes into play. For this purpose, we examine expectation shocks

in t = 0 while all other exogenous parameters are held fixed, in particular all those

related to the second period. Thus, lending is potentially influenced in the following

two ways: First, expectation shocks have a direct effect on the lending decision at

t = 0. Second, the distribution of the bank’s equity at t = 1, WB,1, is affected. As a

consequence, the outcomes of the lending decisions at t = 1 become shifted in a way

that is consistent with the shifts in the distribution of WB,1. So lending decisions at

t = 1 are made as if a realized shock had occurred.

Consequently, the question of whether expectation shocks have a lasting effect is

equivalent to asking the opposite, namely whether expectation shocks at t = 0 can

be fully broken down into a first-period effect that is exclusively based on this original

effect and into a second-period effect that is exclusively based on the realized effect.

If this is true, results from the one-period analysis in Section 6.6 fully carry over to


0

001

002

003

004

005

006

007

561.1551.1541.1531.1521.1511.1501.1590.1

0

001

002

003

004

005

006

007

031.0021.0011.0001.0090.0080.0070.0060.00

001

002

003

004

005

006

007

%5%01%51 %53%03%52%02%51%01%5%

Exp

ecte

d ba

nk e

quity

Correlation between both sector returns in the first periodVolatility of returns on Sector 1 loans in the first period



VaR at the 0.1% level:first period of thetwo-period model


VaR at the 0.1% level:one-period model

Figure 7.3: E0(WB,1) and E(WB) as functions of expectation shocks

This figure illustrates the dependence of the bank’s expected equity E0(WB,1) on expectation shocksin t = 0; that is, when a single parameter of the distribution of gross returns on aggregate loans hasvaried in t = 0 (solid lines with filled-in boxes and diamonds, respectively). As a benchmark, thebank’s expected equity E(WB) is graphed for the one-period case (dashed lines with empty boxes anddiamonds, respectively). The charts represent in counter-clockwise rotation: a shift in the expectedgross return of first-sector loans, µ1,1 (µ1); in the volatility of returns on first-sector loans, σ1,1

(σ1); and in the inter-sectoral return correlation ρ1 (ρ). Expected equity is shown for two differentregimes: the laissez-faire equilibrium and the VaR approach with confidence level p = 0.1%.

the two-period framework. If not, the fact that agents include two periods in their

optimization calculations will influence outcomes economically, thus giving rise to

some kind of accelerator (cf. Bernanke/Gertler/Gilchrist, 1996, 1999). Furthermore,

the bank’s risk-taking behavior may change and in particular be adversely affected

by regulation (Blum, 1999). In particular, Blum (1999) shows that regulation raises

the value of second-period equity. Future equity can only be increased, however, if

a higher risk level is chosen in the first period.

For the analysis, we use the parameter values from Table 6.1, rounded to two decimal

places. Thus, the results concerning the single-period model shown in this chapter

may deviate slightly from the results presented in the previous chapter.


Expe

cted

ban

k eq

uity

0

001

002

003

004

005

006

007

008

009

0001

0011

0021

0031

0041

0051

061.1051.1041.1031.1021.1011.1001.

0

001

002

003

004

005

006

007

008

009

0001

0011

0021

0031

0041

0051

021.0011.0001.0090.0080.0070.00

001

002

003

004

005

006

007

008

009

0001

0011

0021

0031

0041

0051

%53%03%52%02%51%01%5%0%5%01%51


Volatility of returns on Sector 1 loans in the first period Correlation between both sector returns in the first period


VaR at the 0.1% level:first period of thetwo-period model

w/o regulation:second period of thetwo-period model

VaR at the 0.1% level:second period of thetwo-period model

1

Figure 7.4: E0(WB,1) and E0(WB,2) as functions of expectation shocks in t = 0

This figure illustrates the dependence of the bank’s expected equity in t = 1 and t = 2, E0(WB,1) andE0(WB,2), respectively, on expectation shocks at t = 0. The charts represent in counter-clockwiserotation: a shift in the expected gross return of first-sector loans, µ1,1; in the volatility of returnson first-sector loans, σ1,1; and in the inter-sectoral return correlation ρ1. Expected equity is shownfor two different regimes: the laissez-faire equilibrium and the VaR approach with confidence levelp = 0.1%.

7.2.1 Expected Return Shocks

We first vary the expected return on first-sector loans related to the first period,

µ1,1. Figure 7.2 shows the total loan volumes granted by the unregulated and

regulated bank at t = 0 and those granted in the one-period model. Qualitatively, an

expected shock on µ1,1 has the same effects on the sensitivity of total loan volumes

in the first period of the two-period model as in the one-period model (cf. Section

6.6.2). Without regulation, the gap between total loan volumes is striking. As,

by construction, equilibrium magnitudes in t = 0 depend on expected equilibrium

magnitudes related to the second period, this gap ought to be explained by the value

that stems from the possibility of granting loans and generating positive expected

profits in the second period.


0004

0054

0005

0055

0006

0056

0007

0057

561.1551.1541.1531.1521.1511.1the first periodni1rotceS snaolnonruterdetcepxE

Exp

ecte

d to

tal l

oan

volu

me

(w/o

regu

latio

n)

006

056

007

057

008

058

009

Exp

ecte

d to

tal l

oan

volu

me

(VaR

)

w/o regulation:expected values,second period ofthe two-periodmodel

VaR at the0.1% level:expected values,second period ofthe two-periodmodel

501.1

Figure 7.5: Expected total loan volume E0(L1) as a function of µ1,1

This figure illustrates the dependence of the expected total loan volumes on the expected gross returnof first-sector loans, µ1,1. Total loan volumes are shown for two different regimes: the laissez-faireequilibrium and the VaR approach with confidence level p = 0.1%.

In particular, lower total loan volumes result in lower expected equity in t = 1, as

Figure 7.3 shows. But reduced business volume L∗0 in the first period raises the

success probability Φ(b∗dz ,1) and hence, partially increases (assuming that second-

period loan volumes are fixed) equity in t = 2 as expected at t = 0; that is,

E0[W ∗B,2]. Therefore, the bank can raise its continuation value by further offering

its intermediation service in the second period at the expense of expected profits

after the first period. This continuation value can by quantified by the gap

E0[W ∗B,2]− E0[W ∗

B,1], as shown by the upper chart in Figure 7.4.

Under regulation, we cannot confirm that this continuation effect has a significant

adverse effect on risk-taking at t = 0. That is, the total loan volume LV0 granted

in the first period of the two-period model is close to the total loan LV granted

in the one-period model, as Figure 7.2 illustrates. Moreover, the gap between the

respective expected levels of bank equity, E0(W VB,1) and E(W V

B ), do not significantly

differ.2 As a result, the default probabilities Φ(bVdz ,1) and Φ(bVsz) are close to each

other, too.3 Yet, the continuation value, given by E0(WB,2) − E0(WB,1), is clearly

2The relative deviation(

E0(WVB,1)− E(WV

B ))/E0(WV

B,1) ranges between −0.44% and 0.86%for 1.105 ≤ µ1,1 ≤ 1.160 and thus cannot be represented by the respective graphs, as shown in theupper chart of Figure 7.3.

3More precisely, the one-period default probability Φ(bVdz,1), referring to the first period in the


positive, as illustrated by the respective graphs shown in the upper chart of Figure

7.4.

These findings stand in contrast to those obtained by Blum (1999), who shows that

the regulated and thus volume-constrained bank chooses a higher risk level in the

first period in order to raise its expected equity after the first period, as more equity

allows the bank to increase its expected profits in the second period. A major

difference between Blum’s setting and ours is that in his model the risk-neutral

bank is faced with an exogenous deposit cost function that is independent of risk,

whereas in our setting the risk-averse household has a risk-disciplining effect in the

two-period setting as well. Note that the VaR approach did not result in adverse

effects concerning risk-taking in the one-period framework either.

Figure 7.5 shows the expected total loan volume E0[L1(WB,1)] granted in the second

period. It strictly increases in µ1,1 under both the unregulated and the regulated

regimes, although the increase is smaller under regulation. Thus, neither a pro- nor

a counter-cyclical effect on lending exists on average in the second period.

It is still unclear to what extent second-period loan volumes are directly influenced

by shocks in µ1,1 and to what extent they are influenced by the changes in the bank’s

equity WB,1.

We try to answer this question by comparing the reaction of the total loan volume

L ≡ L(WB) to shocks in WB in the one-period framework with the reaction of

expected second-period total loan volumes E0[L1(WB,1)] to the bank’s expected

equity E0[WB,1], where both magnitudes result from the same expectation shock

in µ1,1.

Concerning shocks in µ1,1, this comparison is drawn by means of the upper chart in

Figure 7.6.

The curves representing L∗ and E0[L∗1(W ∗B,1)] are essentially parallel to each other

if L∗ and E0[L∗1(W ∗B,1)] are plotted against E0[W ∗

B,1] on the abscissa. Thus, the

shift in equity levels W ∗B,1 — implied by µ1,1 shocks in the two-period model —

seems to mainly account for the sensitivity of E0[L∗1(W ∗B,1)] with respect to E0[W ∗

B,1].

The parallel distance of E0[L∗1(W ∗B,1)] to L∗(E0[W ∗

B,1]) can be explained by Jensen’s

inequality. In particular, Jensen’s inequality implies E0[L∗1(W ∗B,1)]−L∗(E0[W ∗

B,1]) <

0 since L∗(·) and L∗1(·) are strictly concave functions. As E0[L∗1(W ∗B,1)] and

L∗(E0[W ∗B,1]) are shown in Figure 7.6 as dependent on E0[W ∗

B,1], whereas different

two-period model, strictly decreases from 0.00020% to 0.000023% given expected returns µ1,1 ∈[1.105, 1.160]. Choosing the same range for expected returns within the one-period model yieldsdefault probabilities from 0.00021% to 0.000023%.


Bank’s expected/initial equity

0004

0054

0005

0055

0006

0056

0007

0057

0008

006075045015084054024093063033003

0004

0054

0005

0055

0006

0056

0007

0057

0008

0060750450150840540240930630330030004

0054

0005

0055

0006

0056

0007

0057

0008

006075045015084054024093063033003

1.1601.155

1.145

1.1351.125

1.1151.105

0.1200.100

0.0900.080

0.070

10%5%0%

-10%-5%

30%20%

(Exp

ecte

d) to

tal l

oan

volu

me

1,1

1,1 1,1

w/o regulation:expected value,second period of thetwo-period model


Figure 7.6: (Expected) total loan volume as a function of the bank’s(expected) equity w/o regulation

This figure illustrates the dependence of the expected total loan volume E0(L1) on the bank’sexpected equity E0[WB,1] when an expectation shock has occurred in t = 0 (solid lines with filled-indiamonds). The charts represent in counter-clockwise rotation: a shift in the expected gross returnof first-sector loans, µ1,1; in the volatility of returns on first-sector loans, σ1,1; and in the inter-sectoral return correlation ρ1. As a benchmark, the total loan volume L, as a function of the bank’sinitial equity WB, is graphed for the one-period case (dashed lines with empty diamonds).

levels of E0[W ∗B,1] are identified with different levels of µ1,1, the parallelism of

E0[L∗1(W ∗B,1)] and L∗(E0[W ∗

B,1]) with respect to the E0[W ∗B,1] axis emerges. To make

the comparison meaningful, E0[W ∗B,1] is identified with initial equity WB concerning

the total loan volume L∗(·) in the one-period model. Note, however, that the

functions L∗(·) and L∗1(·) are not identical in general.

While the graphs in Figure 7.6 refer to the laisser-faire equilibrium, Figure 7.7 shows

the (expected) total loan volumes under the VaR-based regulation at a confidence

level of 0.1%.

Concerning µ1,1 shocks, the upper chart of Figure 7.7 illustrates that, under

regulation, the sensitivity of the expected total loan volume granted at t = 1 with

respect to E0[W VB,1] is almost the same as in the one-period model with respect to


Bank’s expected/initial equity

006

036

066

096

027

057

087

018

048

078

009

071561061551051541

006

036

066

096

027

057

087

018

048

078

009

071561061551051541006

036

066

096

027

057

087

018

048

078

009

071561061551051541

1.1601.155

1.1451.135

1.1251.115

1.105

0.070

0.0850.105

0.120

-5%

-10%0%

10%20%

30%

1,1

1,1 1,1

(Exp

ecte

d) to

tal l

oan

volu

me

VaR at the 0.1% level:expected values,second period of thetwo-period model

VaR at the 0.1% level:one-period model

Figure 7.7: (Expected) total loan volume as a function of the bank’s(expected) equity under VaR

This figure illustrates the dependence of the expected total loan volume E0(L1) on the bank’sexpected equity E0[WB,1] when an expectation shock has occurred in t = 0 (solid lines with filled-indiamonds). The charts represent in counter-clockwise rotation: a shift in the expected gross returnof first-sector loans, µ1,1; in the volatility of returns on first-sector loans, σ1,1; and in the inter-sectoral return correlation ρ1. As a benchmark, the total loan volume L, as a function of the bank’sinitial equity WB, is graphed for the one-period case (dashed lines with empty diamonds).

equally high levels of initial equity WB. Thus, the sensitivity of E0[LV1 ] can be mainly

traced back to the shifts in W VB,1 and in E0[W V

B,1], respectively, that are in turn due

to the expectation shock concerning µ1,1 in t = 0. The gap E0[LV1 (W VB,1)]−LV (WB)

almost disappears under regulation, as total volumes are much lower than what they

are without regulation.

Concerning E0[WB,1], the total expected loan volume granted at t = 1 is not

pro-cyclically affected by regulation either: We obtain∆E0[L∗1]

∆E0[W ∗B,1]≈ 5.87 without

regulation versus∆E0[LV1 ]

∆E0[WVB,1]≈ 4.75 under regulation. We note that there are again

regions around lower realized values of WB,1 where the unregulated total loan volume

reacts more strongly to shocks than the total loan volume under VaR regulation,

due to the total loan volume’s stronger concavity without regulation (cf. Fig. 6.3,


0

0001

0002

0003

0004

0005

0006

031.0021.0011.0001.0090.0080.0070.0060.htni1rotceS snaolnosnruterfoytilitaloV

Tota

l loa

n vo

lum

ew/o regulation:first period of thetwo-period model


VaR at the0.1% level:first period of thetwo-period model

VaR at the0.1% level:one-period model

0e first drep io

Figure 7.8: Total loan volume L0 (L) as a function of σ1,1 (σ1)

This figure illustrates the dependence of the equilibrium total loan volumes on the volatility ofreturns on first-sector loans, σ1,1 (σ1). Solid lines represent total loan volumes granted in the firstperiod of the two-period model, while dashed lines indicate total volumes in the one-period model.Total loan volumes are shown for two different regimes: the laissez-faire equilibrium and the VaRapproach with confidence level p = 0.1%.

6.9, and 6.10 in the one-period framework).

7.2.2 Return Volatility Shocks

Figure 7.8 shows the comparative static results for volatility shocks in the first

period. In this example, the volatility of returns on loans granted to Sector 1, σ1,1,

is varied at t = 0. As in the one-period model, total loan volumes decrease strictly in

σ1,1 with and without regulation. In particular, the regulatory constraint becomes

tighter the higher σ1,1 is; this would hold true even if the bank chose the variance-

minimum loan-allocation rate. Regulation does not have a pro-cyclical impact as

demonstrated in the one-period setting (cf. Fig. 6.14 and 6.15).

The total loan volume referring to the first period of the two-period problem is

lower than the equivalent volume in the one-period set-up, partly resulting in default

probabilities Φ(bdz ,1) that are lower than those obtained in the one-period model,

Φ(bsz). These effects observed under σ1,1 shocks are thus similar to those observed

with µ1,1 shocks. The continuation values in terms of E0[WB,2] − E0[WB,1] can be

identified by the graphs, as shown in the bottom-left chart of Figure 7.4. As with


600

650

700

750

800

850

900

4000

4500

5000

5500

6000

6500

7000

7500

0.070 0.080 0.090 0.100 0.110 0.120

Exp

ecte

d to

tal l

oan

volu

me

(VaR

)

Exp

ecte

d to

tal l

oan

volu

me

(w/o

regu

latio

n)

Volatility of returns on Sector 1 loans in the first period



Figure 7.9: Expected total loan volume E0(L1) as a function of σ1,1

This figure illustrates the dependence of the expected total loan volumes on the volatility of returnson first-sector loans, σ1,1. Solid lines represent total loan volumes granted in the first period ofthe two-period model, while dashed lines indicate total volumes in the one-period model. Total loanvolumes are shown for two different regimes: the laissez-faire equilibrium and the VaR approachwith confidence level p = 0.1%.

µ1,1 shocks, the effects on expected equity after the first period, E0[WB,1], compared

to the bank’s expected equity within the one-period setting, E[WB], are negligible,

as the bottom-left chart in Figure 7.3 illustrates.

As shown in Figure 7.9, expected total loan volumes strictly decrease in σ1,1; the

sensitivity is greater for the unregulated volumes than for the regulated volumes.

Hence, there is neither a pro- nor a counter-cyclical effect. Considering the bottom-

left charts in Figures 7.6 and 7.7 reveals that the shift in bank equity WB,1 (implied

by changes in σ1,1) seems to be mainly responsible for the changes in expected loan

volumes in the second period. Hence, there can be no accelerating mechanism with

respect to total loan volume sensitivities, as is also the case with µ1,1 shocks. The

positive sign of the gap L∗(E0[W ∗B,1]) − E0[L∗1(W ∗

B,1)] can be explained again by

Jensen’s inequality, since L∗(·) and L∗1(·) are strictly concave, respectively.


300

400

500

600

700

800

0

1000

2000

3000

4000

5000

6000

-15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35%

Tota

l loa

n vo

lum

e (w

/o re

gula

tion)

Correlation between both sector returns in the first period



VaR at the0.1%-level:first period of thetwo-period model

VaR at the0.1%-level:one-period model

Tota

l loa

n vo

lum

e (V

aR)

Figure 7.10: Total loan volume L0 (L) as a function of ρ1 (ρ)

This figure illustrates the dependence of the equilibrium total loan volumes on the inter-sectoralreturn correlation ρ1 (ρ). Total loan volumes are shown for two different regimes: the laissez-faireequilibrium and the VaR approach with confidence level p = 0.1%.

7.2.3 Return Correlation Shocks

Because correlation shocks concerning ρ1 affect the portfolio return volatility σ1

in the same manner as volatility shocks with respect to a single sector, we expect

similar effects on total loan volumes.

Indeed, the total loan volumes with and without regulation strictly decrease in ρ1

where contractions in unregulated volumes are stronger. Thus, we observe no pro-

cyclical impact caused by regulation, as in the one-period model and as it is the

case with volatility shocks in the one- and in the two-period models.

Again, the decrease in the expected loan volumes E[L1(WB,1)] is comparable to

the effects of equity shocks in the one-period model, as illustrated in each bottom-

right chart of Figures 7.6 and 7.7. We note that the same effects exist concerning

risk-taking in the first period and concerning continuation values, as in both the

preceding comparative-static analyses.


0004

0054

0005

0055

0006

0056

0007

0057

%03%52%02%51%01%5%0%5%0tnisnruterrotceshtobneewtebnoitalerroC

Exp

ecte

d to

tal l

oan

volu

me

(w/o

regu

latio

n)

006

056

007

057

008

058

009

Exp

ecte

d to

tal l

oan

volu

me

(VaR

)



1- -he tri s periodf

Figure 7.11: Expected total loan volume E0(L1) as a function of ρ1

This figure illustrates the dependence of the expected total loan volumes on the inter-sectoral returncorrelation ρ1. Solid lines represent total loan volumes granted in the first period, while dashedlines indicate total volumes in the one-period model. Total loan volumes are shown for two differentregimes: the laissez-faire equilibrium and the VaR approach with confidence level p = 0.1%.

7.3 Summary

Examining two subsequent periods allows us to determine whether expectation

shocks have lasting effects on loan volumes and their sensitivities to these shocks. By

construction, expectation shocks occurring at t = 0 transmit to the second period

via the bank’s equity. Equity is thus an amplifier for expectation shocks, albeit

only if ex-post realizations can be aligned with the ex-ante shifted expectations.

In this respect, we may have identified a financial accelerator in the spirit of

Bernanke/Gertler/Gilchrist (1996, 1999), though other propagation mechanisms,

such as information asymmetries, are missing.

However, there is by no means an acceleration of sensitivities as such. Far from it:

Total loan volumes react to expectation shocks as in the one-period setting when

the very period is considered at whose beginning the shock occurred. Likewise, the

expected or average reaction of total loan volumes in the second period to the shifts

thus implied in the bank’s equity are in the same order of magnitude as in the

one-period model.

The most striking difference between the one-period and the two-period models

7.3. SUMMARY 241

is that total loan volumes in the first period may be well below their single-

period counterparts. The most suitable explanation is that the bank recognizes

its continuation value arising from further intermediation in another period when

making its decisions at t = 0. The importance of continuation values to the bank’s

decision making and risk-taking has been analyzed by Blum (1999). Continuation

values arise in our model due to the bank’s monopoly power, which could be justified

by the bank charters granted by regulatory authorities (Keeley, 1990).


Part IV

Final Remarks

243

Final Remarks

The concern about pro-cyclical movements of aggregate lending volumes through

capital adequacy rules has been debated since the nineties and has attracted

increased attention during the consultation process for the amendments to capital

rules which have become commonly known as the Basel II Accord. With this

thesis, we add to the literature that is concerned with pro-cyclicality in the following

manner: We analyze the sensitivity of the lending volume to changes of fundamental

economic variables, called shocks, under a VaR approach, under approaches with

fixed risk weights, and under a laissez-faire economy. We endogenize the bank’s

risk-allocation and size decision. In particular, the deposit volume and the deposit

interest rate are based on decisions made by the household and the bank. The

deposit interest rate and the deposit volume reflect the bank’s risk-taking and

the regulatory constraints. So far, the literature about pro-cylicality of capital

requirements has not been concerned with a bank that simultaneously takes its

leverage, asset risks, probability of bankruptcy, size, and costs of debt finance into

account. Likewise, there is no work in this respect that analyzes the interaction

between the bank and a risk-averse depositor while endogenizing for the bank’s

bankruptcy risk.

Our results concerning pro-cyclicality lead to the conclusion that the effect of a

given shock should be distinguished according to the type of the shock. Equity

shocks, i.e. changes in bank equity as result of gains or losses, lead to pro-cyclical

effects on lending. Expectation shocks, i.e. changes of distribution parameters or

of borrowers’ productivity, are dampened by regulation. Regulation affects the

total lending volume asymmetrically given each type of shock: Concerning equity

shocks, regulation effectively constrains bank lending after realized losses, thus

exacerbating downturns. This observation is in line with common fears. Concerning

expectation shocks, regulation effectively constrains bank lending in anticipation

of more favorable expectations on outcomes, thus hampering economic recovery.

Except for pro- and non-pro-cyclical effects, we can even observe counter-cyclical

effects through regulation on the level of single loan volumes. These insights can be

245

246

found regardless of the degree of risk sensitivity of the respective capital adequacy

rule. A bank may grant higher loan volumes to less risky firms under risk-sensitive

capital requirements than it would do under fixed requirements or under a laissez-

faire regime.4

The two-period model considered does not yield any further insights relative to the

one-period model. In particular, there are no signs of a financial accelerator that

could be based on a simple propagation mechanism via the bank’s equity. If the

sensitivities of the expected total loan volume in t = 1 toward expectation shocks at

the beginning of the first period, t = 0, are considered as sensitivities with respect

to expected equity in t = 1, these sensitivities are always of the same size as they

are in the one-period model.

Furthermore, this work emphasizes that risk-taking may occur under capital

regulation without any interplay with other regulatory measures, such as deposit

insurance schemes. Rather, the degree of risk-taking can be directly aligned with

the sort of capital rules considered. A flat capital requirement always induces the

bank to take more risk than under any other regime considered (i.e. laissez-faire,

different risk weights, VaR approach). The reduction in size is always compensated

by taking more risk since this is the only possibility to raise the bank’s expected final

wealth. However, as capital requirements become risk-sensitive, our models yield

mixed results. If loan redemptions are normally distributed, the VaR approach

generally has no impact on risk-taking compared to the laissez-faire equilibrium.

Thus, the “correct” alignment of capital rules with actual credit risk is of great

importance, as analyzed by Kim/Santomero (1988) and Rochet (1992). But finding

the “correct” risk weights or risk-determining models will be an unsolvable task in

reality. Therefore, adverse side effects concerning risk-taking will always remain a

problem to various extents. This problem is commonly known as model risk.

The mixed results of this thesis raise the question of the relevancy of the concerns

about pro-cyclical effects through risk-sensitive capital requirements. The research

done on the role of capital buffers suggest that these buffers may alleviate this

problem to a considerable extent (Peura/Jokivuolle, 2004; Jacques, 2005; Heid,

2007, and Repullo/Suarez, 2008). Others, however, who consider pro-cyclicality as

a major problem, propose rules of how pro-cyclicalty could be mitigated. In this vein,

Kashyap/Stein (2004) and Gordy/Howells (2006) suggest counter-cyclical indexing

of capital rules. Pennacchi (2005) advocates a regulatory regime comprising actuary

fair deposit insurance premia and a fixed capital ratio instead of risk-sensitive capital

4Cf. Jokivuolle/Vesala (2007) for a slightly different result.

247

requirements.

This controversy concerning the importance of pro-cyclical impacts may for the most

part exist because of the variety of the frameworks and variables analyzed. First

of all, many authors, including Kashyap/Stein (2004) and Gordy/Howells (2006),

exclusively consider the cyclicality of required capital, thereby assuming that capital

is always completely taken up for lending, such that the existence of buffers is

neglected. This assumption is justified as long as buffers can be seen as a rather

constant endorsement on required capital. This view is also taken in our analysis.

As a consequence, Basel II leads to increased cyclicality of capital compared to Basel

I, in particular with expectation shocks, in their analyses. Consequently, the same

cyclicality is also believed to be true concerning total loan volumes. But this result

does not only differ from the conclusions drawn by the capital-buffer based views,

but also from our reasoning as follows: In their view, swings in capital requirements

are always identified with proportional swings in lending as the bank’s reaction to

changes of risk is not endogenized (apart from ex-ante fixed re-investment rules in

simulation studies). In our analysis, though, the bank reacts differently given the

various regulatory regimes and must account for the re-financing costs associated

with its risk-taking.

Another point might be that the debate on pro-cyclicality often seems to be

related to absolute lending levels, perhaps influenced by the experiences with the

introduction of the Basel I Accord and the credit crunch.

But as the Basel II Accord was introduced in 2008 in the EU and as it has thus taken

effect under the extra-ordinary circumstances of the ongoing financial and economic

crisis, it will take years to fully assess and understand its potential impact on the

business cycle. Furthermore, amendments, known as Basel III, will come into effect

from 2013 on.

248

Appendix A

Proofs to Chapter 3

A.1 The Household

A.1.1 Proof of Result 2

The Existence and Uniqueness of RD(`; j)

To obtain RD(`; j), the equation

Dsj(`, RD)

!= 0

must be solved for RD given ` and j. Multiplying this equality by γ · σ2j and

simplifying reveals that this equation is linear in RD. Thus, RD(`; j) is unique and

is equal to (3.2.55) after having solved this equation under consideration of (3.2.53).

The Existence and Uniqueness of Ru

D(`; j)

By the implicit function theorem the derivative of the unconstrained deposit-supply

function with respect to RD is given by

dDsj

dRD

= −∂2Uj(D)

∂D∂RD∂2Uj(D)

∂D2

.

Because of∂2Uj()

∂D2 = −γσ2j < 0, the sign of this derivative is exclusively determined

by the sign of the second mixed derivative

∂2Uj(D)

∂D∂RD

= qj − γ ·[2qj(RD − µj) ·D − qj ·Rµ

j ·WB

](A.1.1)

249

250 APPENDIX A. PROOFS TO CHAPTER 3

which is positive if and only if

RD <

1γ

+ 2µjD +Rµj ·WB

2D

holds. Let

fj(RD) =

1γ

+ 2µjD +Rµj ·WB

2D.

If RD approaches the zero RD(`; j) from above, fj(RD) becomes infinite,

limRD↓RD(`;j)

fj(RD) = +∞ . (A.1.2)

Its derivative is given by

f ′j(RD) =1

2γD2·[2γqjD

2 − (1 +Rµj ·WB) · ∂D

uj

∂RD

]. (A.1.3)

Thus, the following relation between the signs of∂Duj∂RD

and f ′j(RD) can be established:

∂Duj∂RD

< 0 ⇒ f ′j(RD) > 0

f ′j(RD) < 0 ⇒ ∂Duj∂RD

> 0. (A.1.4)

At RD = RD(`; j), the deposit-supply function is strictly increasing due to

∂2Uj(D)

∂D∂RD

∣∣∣∣RD=RD(`;j)

= qj + qj ·Rµj · γWB > 0 . (A.1.5)

As long as fj(RD) > RD holds for RD > RD(`; j), the deposit-supply function keeps

increasing in RD. Furthermore, fj(RD) is decreasing in RD in the neighborhood of

RD(`; j) due to

limRD↓RD(`;j)

f ′j(RD) = −∞ .

With increasing RD, fj(RD) could either run such that it crosses the identity RD ≡RD at least once, or run such that it never crosses the identity. Since the latter

case is associated with fj(RD) > RD for all RD, the deposit-supply function keeps

increasing for all RD. This contradicts the fact that

limRD→∞

Ds(`, RD; j) = 0+ (A.1.6)

holds, implying that a critical rate RD exists from which the deposit-supply function

decreases in RD. Hence, the function fj(RD) crosses at least once the identity

line. As soon as RD > fj(RD) holds, the deposit-supply function is decreasing.

A.1. THE HOUSEHOLD 251

RD = RDf1 (RD)

1.2

1.4

1.6

1.8

2.0

1.2 1.4 1.6 1.8 2.0

0.00

0.01

0.02

0.03

0.04

- 0.01 1.2 1.4 1.6 1.8 2.0

f1 (RD) - RD

Figure A.1: The function f1(RD) and its relation to the identity RD ≡ RD

This figure illustrates the course of the function f1(RD). The left-hand chart shows the graph off1(RD) with the parameter values given in Table 4.1 and given ` = 0.57114. The right-hand chartshows the graph of f1(RD)−RD with the same parameter values to highlight the unique intersectionpoint. The chosen loan-allocation rate equals its respective laissez-faire equilibrium value.

By (A.1.4), the function fj(RD) starts increasing after this intersection. Hence, if

fj(RD) now crosses the identity another time, it crosses the identity from below.

Therefore, f ′j(RD) > 1 must hold in a potential second intersection point. Because

of fj(RD) = RD in any intersection point, the deposit supply function has zero slope

by the definition of fj(RD) and f ′j(RD) becomes f ′j(RD) = qj < 1.

Thus, fj(RD) keeps bounded by RD from the first intersection point on and

the deposit-supply function decreases in RD. So, the intersection is unique and,

moreover, the deposit-supply function is maximized in that point with respect to

RD. Thus, the intersection point is equivalent to Ru

D(`) defined in (3.2.52).

Under Case 4, the behavior of the deposit-supply function concerning RD can also

be shown by taking the derivative of the explicit function Du(`, RD; 4) with respect

to RD.

Figure A.1 displays the graph of f1(RD) given the parameter values from Table

4.1 to illustrate the idea of the proof. To highlight the unique intersection point,

f1(RD) − RD is shown as well since this difference is only roughly one hundredth

compared to the lengths of the intervals considered as ranges.

A.1.2 Derivations concerning Result 3

Existence and uniqueness of RD(`; j) is given by Result 2.


To obtain RD(`; j), the equation

Ds(`, RD; j)!

= 0

must be solved for RD given ` and j. Multiplying this equality by γ · σ2j and

simplifying leads to (3.2.55) after having solved this equation under consideration

of (3.2.53).

The derivative of RD(`; j) is given by

∂RD(`; j)

∂WB

=γ ·[Rσj − Rf ·Rµ

j

]

qj · (1 + γWBRµj )2

.

Consider Case 2. The derivative becomes

∂RD(2)

∂WB

=γ · (p1 − q)α1` [α1` − Rf ]

p2 · (1 + γWBRµ2 )2

.

By (3.2.20), we obtain

∂RD(2)

∂WB

≤γ · (p1 − q)α1`

[(p1 − q) · α1α2

α1+α2− Rf

]

p2 · (1 + γWBRµ2 )2

.

By (3.2.7), we have α1α2

α1+α2≤ 1 and due to (3.2.6), we finally obtain

∂RD(2)

∂WB

≤ 0 .

Under Case 3 we can derive zero as an upper bound in an analogical way.

If WB = 0, RD(`; j) becomes

RD(`; j) =Rf −Rµ

j

qj.

In Case 2, we obtain:

Rf −Rµ2

q2

=Rf − (p1 − q)α1`

p2

.

This term strictly exceeds Rf iff

(1− p2)Rf > (p1 − q)α1`

A.1. THE HOUSEHOLD 253

is fulfilled. By (3.2.6), (1− p2)Rf ≥ 1− p2 holds, and by (3.2.7) as well as (3.2.20)

we obtain (p1 − q)α1` ≤ (p1 − q) α1α2

α1+α2≤ p1 − q as higher bound on the right-hand

side. Taking together this bound and the lower of the left-hand side, we obtain

(1− p2)Rf ≥ 1− p2 > p1 − q ≥ (p1 − q)α1`

and 1−p2 > p1−q is true because of (3.2.11). For Case 3 we can argue analogously.

Consider Case 1, we obtain:

Rf −Rµ1

q1

=Rf − (p1 − q)α1`− (p2 − q)α2(1− `)

q,

so that

(1− q)Rf > (p1 − q)α1` + (p2 − q)α2(1− `)

must be fulfilled. Taking the (sufficient) condition stated in the result, the right-

hand side is bounded from above by

(p1 − q)α1`+ (p2 − q)α2(1− `) < (1− q)Rf`+ (1− q)Rf (1− `) = (1− q)Rf .

As this bound equals the left-hand side, we have shown the result.

A.1.3 Proof of Result 4

Proof. Throughout this proof, we will consider the unconstrained deposit-supply

function.

Let us start with Case 2. Deposit supply can be written also as

Du(`, RD; 2) =µ2 −Rf

γσ22

+(p1 − q)α1`(µ2 − α1`)

σ22

·WB .

Hence, the sign of

(p1 − q) · α1` · (µ2 − α1`)

must be determined. By (3.2.20) we can establish as a lower bound

(p1 − q) · α1` · (µ2 − α1`) ≥ (p1 − q) · α1` ·(µ2 −

α1 · α1

α1 + α2

).

By α1, α2 ≤ 2 according to (3.2.7), we obtain α1·α1

α1+α2≤ 1. By the assumption µj ≥ Rf

and by Rf ≥ 1, the result obtains.


Given Case 3 the proof is analogous: We must determine the sign of

(p2 − q) · α2(1− `) · [µ3 − α2(1− `)] .

By (3.2.21), µ3 − α2(1− `) is bounded from below as follows:

µ3 − α2(1− `) ≥ µ3 − α2 ·(

1− α2

α1 + α2

)≥ µ2 − 1

whereas the last inequality is due to (3.2.7). Assumption µj ≥ Rf leads to the

result.

A.2 Results on Specific Instances of the Equilib-

rium

A.2.1 The Equilibrium without Regulation

A.2.1.1 Proof of Result 5

Proof. The bank’s objective function E[WB(`, RD; j)] is a composition of continuous

functions. In particular, Ds(`, RD; j) is continuous (but not necessarily differentiable

as it can be cut off by the household’s initial wealth WH at some points, that

is especially in RD(`; j), RD(`; j) < Ru

D(`; j), for fixed loan-allocation rates).

Furthermore, feasible loan-allocation rates ` are restricted to the compact sets

[0, 1]∩Cj, j = 1, . . . , 4 for fixed deposit rates RD by definition. Note that the sets Cj

are compact in ` for given RD as their Definitions (3.2.16) to (3.2.19) always contain

their respective borders ∂Cj. In particular, the loan-allocation rate is restricted

to (3.2.20) under Case 2 and (3.2.21) under Case 3, respectively. Thus, there is

always at least one loan-allocation rate `∗ = `(RD) maximizing the bank’s objective

function for given deposit interest rates RD. This argument holds for each Case j

as well as globally across all four Cases.

Concerning RD we will argue case-by-case. Note that the bank’s objective function

is differentiable given each of the Cases j = 1, . . . , 4. Let us start with

Case 1:

The partial derivative of the bank’s objective function is given by

∂E[WB(`,RD;1)]∂RD

= −q ·Du(`, RD; 1) + q · [α1`+ α2(1− `)−RD] · ∂Du(`,RD;1)∂RD

A.2. RESULTS ON SPECIFIC INSTANCES OF THE EQUILIBRIUM 255

whereas the household is assumed to be unconstrained. By Result 2 this derivative

takes the following values in RD(`; j), and Ru

D(`; j), respectively:


∣∣∣RD=RD(`;1)

= q · [α1`+ α2(1− `)−RD(`; 1)] ∂Du(`,RD;1)∂RD

∣∣∣RD=RD(`;1)

> 0


∣∣∣RD=R

uD(`;1)

= −q ·Du(`, Ru

D(`; 1); 1) < 0 .

Thus, the derivative ∂E[WB(`,RD;1)]∂RD

changes signs at least once on (RD(`; 1), Ru

D(`; 1))

and hence there is at least one local maximum on (RD(`; 1), Ru

D(`; 1)). Either if

Ru

D(`; 1) > α1`+ α2(1− `) or Ru

D(`; 1) ≤ α1`+ α2(1− `) holds, the expression

q · [α1`+ α2(1− `)−RD] · ∂Du(`, RD; 1)

∂RD

is always negative on (α1`+ α2(1− `), Ru

D(`; 1)), and on (Ru

D(`; 1), α1`+ α2(1− `)),respectively. Note that in the first case the marginal profit [α1` + α2(1 − `) − RD]

is negative, whereas the derivative of the deposit supply is positive due to Result 2

and that in the latter case signs are reversed.

Beyond RD > α1`+ α2(1− `), the contribution of collecting deposits to the bank’s

expected final wealth is negative. Furthermore, the bank’s expected final wealth is

increasing at RD = RD(`; 1). Thus, there is at least one maximizing interest rate

RD on (RD(`; 1), Ru

D(`; 1)). If deposit supply is effectively cut off by the household’s

expected wealth at a rate RD(`; 1) < Ru

D(`; 1), the optimal RD is bounded from

above by RD(`; 1). In this case, the derivative ∂E[WB(`,RD;1)]∂RD

is negative for all those

RD > RD(`; 1) until the bank’s expected final wealth turns negative, such as in the

case of an unconstrained deposit supply.

Concerning uniqueness, consider again


= −q ·Du(`, RD; 1) + q · [α1`+ α2(1− `)−RD] · ∂Du(`,RD;1)∂RD

,

using the implicit function theorem for representing the partial derivative ∂Du(`,RD;1)∂RD

,

this partial derivative of the bank’s objective function becomes

− q ·Du(`, RD; 1) +

+ q · [α1`+ α2(1− `)−RD] · q· 1−γ[2·(RD−µ1)·Du(`,RD;1)−Rµ1 ·WB] γσ2

1



RD <−γσ2

1Du(`,RD;1)+[α1`+α2(1−`)−RD]·[1+γWBR

µ1 +2µ1γDu(`,RD;1)]

2γDu(`,RD;1)·[α1`+α2(1−`)−RD]

holds. For simplicity, Du(`, RD; 1) is rephrased by D. Let

g1(RD) =−σ2

1D + [α1`+ α2(1− `)−RD] ·[

1γ

+ 2µ1D +WB ·Rµ1

]

2D · [α1`+ α2(1− `)−RD]

≡ − σ21

2 · [α1`+ α2(1− `)−RD]+ f1(RD)

If RD approaches the zero RD(`; 1) from above, g1(RD) becomes infinite,

limRD↓RD(`;1)

g1(RD) = − σ21

2 · [α1`+ α2(1− `)−RD(`; 1)]+ lim

RD↓RD(`;1)f1(RD) = +∞ .

(A.2.1)


g′1(RD) = − σ21

[α1`+ α2(1− `)−RD]2+ f ′1(RD) (A.2.2)

where f ′1(RD) is given according to (A.2.2). So, we obtain

limRD↓RD(`;1)

g′1(RD) = −∞ .

As the function g1(RD) emerges from f1(RD) by a variable shift, g1(RD) can be

discussed relative to the better-known function f1(RD). On ( RD(`; 1), α1`+α2(1−`) ), the gap

f1(RD)− g1(RD) =σ2

1

2 · [α1`+ α2(1− `)−RD]> 0

is monotonically increasing until it becomes infinitely large as RD approaches α1`+

α2(1− `) from below. The fact that g1(RD) is bounded by f1(RD) which crosses the

identity RD ≡ RD exactly once on ( RD(`; 1), α1`+ α2(1− `) ) and whose distance

to g1(RD) is increasing in RD, implies that the function g1(RD) crosses identity

RD ≡ RD exactly once, too. This unique intersection point is R∗D by the definition

of g1(RD).

However, this analysis is only true, if a solution such as this is neither restricted by

the Case constraints nor by the depositor’s wealth WH .

The Case constraints may require the bank to choose RD > α1`+ α2(1− `) given `


fixed. Because of Ds(`, RD; 1) > 0, any such choice is dominated by autarky, i.e. by

choosing D∗ = 0 yielding p2α2WB because of (3.2.5) and the bank’s risk neutrality.

This choice, however, is in turn dominated by choosing ` = 0 under Case 2 for all

feasible parameter values as

(p2α2 − p2RD) · p2RD −Rf

γp2(1− p2)R2D

+ p2α2WB > p2α2WB

holds for 1p2Rf < RD < α2. Due to (3.2.5) and (3.2.6), this double inequality may

be fulfilled and in particular R∗D > 1p2Rf ≡ RD(0; 2) holds.

Consider the other possibility that the depositor is effectively constrained by its

initial wealth: If RD(`; 1) is greater than the intersection point satisfying RD ≡g(RD), nothing changes. If RD(`; 1) is lower than that intersection point, the bank’s

objective function E[WB(`, RD; 1)] is still strictly increasing in RD as RD < g(RD)

holds below that intersection. Thus, R∗D = RD(`; 1) is optimal. As RD(`; 1) ≡Ru

D(`; 1) if the depositor is not effectively constrained, we finally obtain that R∗D is

uniquely from ( RD(`; 1), RD(`; 1) ].

Next, we consider

Case 2:

The proof for this Case is analogous to that for Case 1. First, let us consider the

partial derivative of the bank’s objective function


= −p2 ·Du(`, RD; 2) + [qα1`+ p2α2(1− `)− p2RD] · ∂Du(`,RD;2)∂RD

whereas the household is assumed to be unconstrained. By Result 2 this derivative

takes the following values in RD(`; 2), and Ru

D(`; 2), respectively:


∣∣∣RD=RD(`;2)

= [qα1`+ p2α2(1− `)− p2RD(`; 2)] · ∂Du(·,·;2)∂RD

∣∣∣RD=RD(`;2)

> 0


∣∣∣RD=R

uD(`;2)

= −p2 ·Du2 (`, R

u

D(`; 2); 2) < 0 .

Thus, the derivative ∂E[WB(`,RD;2)]∂RD

changes signs at least once on (RD(`; 2), Ru

D(`; 2))

and hence there is at least one local maximum on (RD(`; 2), Ru

D(`; 2)). Either if

Ru

D(`; 2) > qp2α1`+ α2(1− `) or R

u

D(`; 2) ≤ qp2α1`+ α2(1− `) holds, the expression

[qα1`+ p2α2(1− `)− p2RD(`; 2)] · ∂Du(`, RD; 2)

∂RD


is always negative on ( qp2α1`+α2(1−`), Ru

D(`; 2)), and on (Ru

D(`; 2), qp2α1`+α2(1−`)),

respectively. Note that in the first case the marginal profit [qα1` + p2α2(1 − `) −p2RD(`; 2)] is negative, whereas the derivative of the deposit supply is positive due

to Result 2 and that in the latter case signs are reversed.

Since beyond RD > qp2α1`+ α2(1− `), the bank’s expected final wealth is negative,

E[WB(`, RD; 2)] < 0, and since the bank’s expected final wealth is increasing atRD =

RD(`; 2), there is at least one maximizing interest rate RD on (RD(`; 2), Ru

D(`; 2)).

If deposit supply is effectively cut off by the household’s expected wealth at a rate

RD(`; 2) < Ru

D(`; 2), the optimal RD is bounded from above by RD(`; 2). Then

the derivative ∂E[WB(`,RD;2)]∂RD

is negative for all those RD > RD(`; 2) until the bank’s

expected final wealth turns negative, such as in the case of an unconstrained deposit

supply.

Concerning uniqueness, consider again


= −q ·Du(`, RD; 2) + [qα1`+ p2α2(1− `)− p2RD] · ∂Du(`,RD;2)∂RD

,

using the implicit function theorem for representing the partial derivative ∂Du(`,RD;2)∂RD

,

this partial derivative of the bank’s objective function becomes

− p2 ·Du(`, RD; 2) +

+ p2 · [qα1`+ p2α2(1− `)− p2RD] · p2· 1−γ[2·(RD−µ2)·Du(`,RD;2)−Rµ2 ·WB] γσ2

2


RD <−γσ2

2Du(`,RD;2)+[qα1`+p2α2(1−`)−p2RD]·[1+γWBR

µ2 +2µ2γDu(`,RD;2)]

2γDu(`,RD;2)·[qα1`+p2α2(1−`)−p2RD]

holds. For simplicity, Du(`, RD; 2) is rephrased by D. Let

g2(RD) =−σ2

2D + [α1`+ p2α2(1− `)− p2RD] ·[

1γ

+ 2µ2D +WB ·Rµ2

]

2D · [α1`+ p2α2(1− `)− p2RD]

≡ − σ22

2 · [α1`+ α2(1− `)−RD]+ f2(RD)

If RD approaches RD(`; 2) from above, g2(RD) becomes infinite,

limRD↓RD(`;2)

g2(RD) =

= − σ22

2 · [qα1`+ p2α2(1− `)− p2RD(`; 2)]+ lim

RD↓RD(`;2)f2(RD) = +∞ .



g′2(RD) = − σ22

[qα1`+ p2α2(1− `)− p2RD]2+ f ′2(RD) (A.2.3)

where f ′2(RD) is given according to (A.2.3). So, we obtain

limRD↓RD(`;2)

g′2(RD) = −∞ .

As the function g2(RD) emerges from f2(RD) by a variable shift, g2(RD) can be

discussed relative to the better-known function f2(RD). On ( RD(`; 2), qp2α1` +

α2(1− `) ), the gap

f2(RD)− g2(RD) =σ2

2

2 · [qα1`+ p2α2(1− `)− p2RD]> 0

is monotonically increasing until it becomes infinitely large as RD approaches qp2α1`+

α2(1− `) from below. The fact that g2(RD) is bounded by f2(RD) which crosses the

identity RD ≡ RD exactly once on ( RD(`; 2), qp2α1`+α2(1− `) ) and whose distance

to g2(RD) is increasing in RD, implies that the function g2(RD) crosses the identity

RD ≡ RD exactly once, too. This unique intersection point is R∗D by the definition

of g2(RD). Finally, we can argue such as in Case 1 when deposit supply is effectively

bounded by the depositor’s initial wealth WH .

The proofs for the Cases 3 and 4 are analogous. Case-dependent expressions must

be appropriately rephrased. The equilibrium given Case 4 can be determined

analytically so that a proof by indirect arguments is not necessary.


Proof. Let us consider the following simplified version of the Maximization Problem

(3.3.1) given Case 1,

max`, RD

E[WB(`, RD; 1)

],

and let us assume that the household is not constrained by its initial wealth WH .

The associated first-order conditions are:

q · (α−RD) · ∂Du(`, RD; 1)

∂`!

= 0 (A.2.4)

q · (α−RD) · ∂Du(`, RD; 1)

∂RD

− q ·Du(`, RD; 1)!

= 0 . (A.2.5)


By the first first-order constraint,

R∗D = α or∂Du(`, RD; 1)

∂`= 0

must hold. Consider first R∗D = α. Then the bank’s expected wealth is only qαWB

as it forgoes any profit opportunities from intermediation. As

Du(`, α, 1) =pα−Rf − (1− 2`+ 2`2 − p)(p− q)α2γWB

[1− 2`(1− `)] p(1− p) + 2`(1− `)(q − p2) · α2 · γ

is strictly positive for example at ` = 12. And as it is still so for RD < α, the bank

could lessen the deposit interest rate resulting in strictly positive expected wealth

from doing intermediation. Thus, R∗D = α contradicts optimality. Consequently,∂Du(`,RD;1)

∂`= 0 holds. By

∂Du(`, RD; 1)

∂`=

∂Du(`, RD; 1)

∂σ21

· ∂σ21

∂`,

` = 12

solves the first first-order condition. Because of

∂2Du(`, RD; 1)

∂`2=

∂2Du(`, RD; 1)

∂(σ21)2

·(∂σ2

1

∂`

)2

+∂Du(`, RD; 1)

∂σ21

· ∂2σ2

1

∂`2,

the solution ` = 12

is a local maximum as

∂2Du(`, RD; 1)

∂`2

∣∣∣∣`= 1

2

=∂Du(`, RD; 1)

∂σ21︸︷︷︸

< 0

· ∂2σ2

1

∂`2︸︷︷︸> 0

< 0 (A.2.6)

holds. As it is the only critical point independent of RD, ` = 12

maximizes uniquely

the objective E[WB(`, RD; 1)]. By Result 5, the optimal R∗D is unique. In particular,

due to

Du(12, RD; 1) =

qRD+(p−q)α−Rf+(p−q)[qRD+(p−q)α−α 12 ]αγWB

σ21 ·γ

> 0

the relation∂Du(1

2, RD; 1)

∂RD

> 0

holds. For the positive sign of [qRD+(p−q)α−α 12] we refer to Appendix A.2.1.8.



Proof. The sensitivity of the optimal deposit volume with respect to WB is given by

dD∗

dWB

=∂Du(·)∂RD

· dR∗D

dWB

+∂Du(·)∂WB

as∂Du(·)∂`

∣∣∣∣`= 1

2

= 0 (A.2.7)

holds in optimum (cf. Appendix A.2.1.2). The sensitivities of the optimal loan-

allocation rate `∗ and R∗D with respect to WB are given by

(d`∗

dWBdR∗DdWB

)= − 1

det( H(E[·],`∗,R∗D) ) ·(

∂2E[·]∂R2

D− ∂2E[·]∂`∂RD

− ∂2E[·]∂`∂RD

∂2E[·]∂`2

)·(

∂2E[·]∂`∂WB∂2E[·]

∂RD∂WB

)

where H(f, x) denotes the Hessian of function f evaluated at x. By

∂2E[WB(`, RD; 1)]

∂`∂RD

∣∣∣∣∣`= 1

2

= 0 and∂2E[WB(`, RD; 1)]

∂`∂WB

∣∣∣∣∣`= 1

2

= 0

the sensitivities simplify to

(d`∗

dWBdR∗DdWB

)= − 1

det( H(E[·],`∗,R∗D) ) ·(

0∂2E[·]∂`2· ∂2E[·]∂RD∂WB

).

As (`∗, R∗D) is an inner optimum to the maximization problem, the Hessian

H(E[·], `∗, R∗D) is negative semidefinite (cf. Mas-Colell/Winston/Green, 1995,

p. 955). The Hessian itself is simply given by

H(E[·], `∗, R∗D) =

(∂2E[·]∂R2

D0

0 ∂2E[·]∂`2

),

where∂2E[WB(`, RD; 1)]

∂`2

∣∣∣∣∣`= 1

2

< 0

can be derived from (A.2.4) and (A.2.6), respectively. By Result 6, the bank’s

objective function cannot be flat around R∗D either. Hence,

∂2E[WB(`, RD; 1)]

∂R2D

∣∣∣∣∣`= 1

2,RD=R∗D

< 0


holds. Thus, H(E[·], `∗, R∗D) is even negative definite, and its determinant is strictly

positive, det (H(E[·], `∗, R∗D)) > 0. So the sensitivities are well-defined. In total, the

sign ofdR∗DdWB

arises out of the implicit function theorem as follows:

dR∗DdWB

= −

> 0︷︸︸︷1

det ( H(E[·], `∗, R∗D) )·

< 0︷︸︸︷∂2E[·]∂`2

︸︷︷︸> 0

· ∂2E[·]∂RD∂WB

,

which can be reduced to

dR∗DdWB

= −∂2E[·]

∂RD∂WB

∂2E[·]∂R2

D

,

due to det (H(E[·], `∗, R∗D)) = ∂2E[·]∂R2

D· ∂2E[·]

∂`2.

The second mixed derivative of E[·] with respect to RD and WB reads

∂2E[WB(`, RD; 1)]

∂RD∂WB

= − q · ∂Du(`, RD; 1)

∂WB

+ q · (α−RD) · ∂2Du(`, RD; 1)

∂RD∂WB

resulting in the total derivative of the deposit volume with respect to WB as follows:

dD∗

dWB

=∂Du(·)∂RD

· dR∗D

dWB

+∂Du(·)∂WB

= − ∂Du(·)∂RD

·− ∂Du(·)

∂WB+ (α−RD) · ∂2Du(·)

∂RD∂WB

− 2 · ∂Du(·)∂RD

+ (α−RD) · ∂2Du(·)∂R2

D

+∂Du(·)∂WB

.

As −∂2E[·]∂R2

D> 0, we can multiply both sides of the equality by −∂2E[·]

∂R2D> 0 without

changing either side’s sign. As a consequence, the total derivative dD∗

dWBis positive if

and only if

∂Du(·)∂RD

· ∂Du(·)

∂WB

+ (α−RD) ·[∂2Du(·)∂RD∂WB

· ∂Du(·)

∂RD

− ∂2Du(·)∂R2

D

· ∂Du(·)

∂WB

]> 0

holds. In the optimum (`∗, R∗D),

∂Du(12, RD; 1)

∂RD

> 0 and∂Du(1

2, RD; 1)

∂WB

> 0

are satisfied, cf. Appendices A.2.1.2 and A.2.1.8, respectively. Furthermore, the

(unconstrained) deposit-supply function is strictly concave with respect to RD: The


second derivative of the deposit supply function with respect to RD reads

∂2Du(·)∂R2

D

∣∣∣`= 1

2

=

8q−α3(p−q)2[2q(p−q)+2p+q−1]+2(1−q)2qR2D(3Rf−qRD)−6α(p−q)(1−q)qRD[2Rf+(1−q)RD]

γ−(p−q)(1−2p+2q)α2+4q(p−q)αRD−2(1−q)qR2D3

− 8(p−q)qα2−6q3RD+Rf−2pRf+q2[(9+6p)RD+6Rf ]+q[−3(1+2p)RD+Rf−6pRf ]γ−(p−q)(1−2p+2q)α2+4q(p−q)αRD−2(1−q)qR2

D3

− 4(p−q)2qα4·(p−q)[4(p−q)(1+q)−3]+(1−p)WB

−(p−q)(1−2p+2q)α2+4q(p−q)αRD−2(1−q)qR2D3

+4(p−q)q2(1−q)αRD·6(1−2p+2q)[(1−q)RD−(p−q)α]α−4(1−q)qR2

DWB

−(p−q)(1−2p+2q)α2+4q(p−q)αRD−2(1−q)qR2D3

.

Numerical routines implemented in Mathematica show that the numerator is greater

than zero on the domain specified by

(p, q, α,RD, Rf , γ,WB) : 2p− 1 ≤ q ≤ p , 12≤ p ≤ 1, pα > RD > Rf ≥ 1,

α ≤ 2, γ ≥ 0, WB ≥ 0 .

The denominator, however, becomes negative on the same domain. More specifically,

the denominator takes all values from [−0.0838413, 0). Thus, we obtain

∂2Du(`, RD; 1)

∂R2D

∣∣∣∣`= 1

2

< 0

where all parameter values have not been more restricted than already done. Hence,

the problem of determining the sign of dD∗

dWBcan be broken down as follows:

∂Du(·)∂RD

· ∂Du(·)

∂WB︸︷︷︸>0

+ (α−RD)︸︷︷︸>0

·[∂2Du(·)∂RD∂WB

· ∂Du(·)

∂RD

︸︷︷︸sign?

− ∂2Du(·)∂R2

D

· ∂Du(·)

∂WB

]

︸︷︷︸>0

> 0

By ∂Du(·)∂RD

> 0, it is now sufficient to show that

∂Du(·)∂WB

+ (α−RD) · ∂2Du(·)

∂RD∂WB

> 0


holds. But the sign of

∂2Du(12, RD; 1)

∂RD∂WB

=−2(p−q)qα[(p−q)(1−2p+2q)α2−2(1−2p+2q)(1−q)αRD+2(1−q)qR2

D]

[(p−q)(1−2p+2q)α2−4(p−q)qαRD+2(1−q)qR2D]

2

is ambiguous. At `∗ = 12, the expression ∂Du(·)

∂WB+ (α−RD) · ∂2Du(·)

∂RD∂WBcan be displayed

as

− (p−q)α(p−q)[(1+2q)(1−4(p−q))+4p(p−q)]α3+4q[2(p−q)(2p−3q)−(1−q)]α2RD[(p−q)(1−2p+2q)α2−4(p−q)qαRD+2(1−q)qR2

D]2

− (p−q)α2q[3(1−2p+q)+2q(1−q+5(p−q))]αR2D−8(1−q)q2R3

D[(p−q)(1−2p+2q)α2−4(p−q)qαRD+2(1−q)qR2

D]2 .

The numerator except for the multiplier (p− q)α takes values from [0, 0.238193] on

the domain specified by

(p, q, α,RD, Rf , γ,WB) : 2p− 1 ≤ q ≤ p , 12≤ p ≤ 1, pα > RD > Rf ≥ 1,

α ≤ 43, γ ≥ 0, WB ≥ 0

.

Note that the firms’ return parameter α has been further restricted as otherwise

negative values become possible. Thus, α ≤ 43

is an additional (compared to the

assumptions made so far) and sufficient condition to ensure that

dD∗

dWB

> 0

holds.


Proof. Solving the first-order condition given Case 2

∂E[·]∂RD

∣∣∣∣`=0

= −p2 ·Du(·; 2) + [qα1`+ p2α2(1− `)− p2RD] · ∂Du(·; 2)

∂RD

!= 0 (A.2.8)

for RD after having plugged-in ` = 0 results readily in

R∗D =2α2Rf

p2α2 +Rf

(A.2.9)


being the unique maximizer according to Result 5. Straight-forward calculations

show that the optimal deposit volume becomes

Du(0, R∗D, 2) =(p2α2 −Rf ) · (p2α2 +Rf )

4p2(1− p2)α22Rfγ

and the bank expects

E[WB(0, R∗D, 4)

]=

(p2α2 −Rf )2

4(1− p2)α2Rfγ+ p2α2WB

as final wealth. A necessary and sufficient condition for `∗ = 0 as optimum is that

∂E[·]∂`

∣∣∣`=0,RD=R∗D

= (qα1 − p2α2) · (Du(·, 2) +WB) +

+ [qα1`+ p2α2(1− `)− p2RD] · ∂Du(·,2)∂`

= (qα1 − p2α2) · (Du(·, 2) +WB) + p2 · (α2 −R∗D) · ∂Du(·,2)∂`

!

≤ 0

holds as we have assumed that there is a unique global maximizer with respect to

the loan-allocation rate ` on the reals, R. This condition can be stated in terms of

the parameters of the model as

−(1− p2)p2α2Rf + [(p1 − q)p2α2 + (q − p1p2)Rf ] · α1 +

+ 2p2(1− p2)α22 · γWB ·Rf ·

· (2− p2)(qα1 − p2α2)Rf + p2(p2α2 − p1α1)Rf + p22(p1 − q)α1α2

!

≤ 0

such that we obtain Condition (3.3.3).

If this condition holds with equality, `∗ = 0 is the maximizer of E[WB(`, ·, 2)] with

respect to the whole real line R. Here, we will refer to it as the unconstrained case.

If, however, Condition (3.3.3) strictly holds, `∗ = 0 is the corner solution to the

maximization problem where the loan-allocation rate is restricted to [0, 1]. We refer

here to this case as the constrained case. First, let us consider

The Unconstrained Problem

which implies that the sensitivities of `∗ and R∗D can be calculated according to the

implicit function theorem:

(d`∗

dWBdR∗DdWB

)= − 1

det( H(E[·],`∗,R∗D) ) ·(

∂2E[·]∂R2

D− ∂2E[·]∂`∂RD

− ∂2E[·]∂`∂RD

∂2E[·]∂`2

)·(

∂2E[·]∂`∂WB∂2E[·]

∂RD∂WB

)


where H(f, x) denotes the Hessian of function f evaluated at x. The notation E[·]should be understood here as short-hand for the bank’s expected final wealth given

Case 2 and its respective derivatives as being evaluated at the optimum (`∗, R∗D).

At this point, the second mixed derivative

∂2E[·]∂RD∂WB

∣∣∣`=0,RD=R∗D

= −p2 · ∂Du(·,2)∂WB

+ [qα1`+ p2α2(1− `)− p2RD] · ∂2Du(·,2)∂RD∂WB

vanishes as both partial derivatives of the deposit-supply function vanish

(cf. Formula (3.2.48) in Result 1) with respect to the derivative ∂Du(·,2)∂WB

) such that

the sensitivities simplify to

(d`∗

dWBdR∗DdWB

)= − 1

det( H(E[·],`∗,R∗D) ) ·(

∂2E[·]∂R2

D· ∂2E[·]∂`∂WB

− ∂2E[·]∂`∂RD

· ∂2E[·]∂`∂WB

)(A.2.10)

As the maximum refers to the unconstrained problem (i.e. the bank is allowed

to short sell) the determinant of the Hessian is positive (since the Hessian of

E[WB(`, RD; 2)] is a 2 × 2-matrix and (`∗, R∗D) = (0,2α2Rfp2α2+Rf

) is assumed to

be the maximizer of the problem). The relevant derivatives, all evaluated at

(`∗, R∗D) = (0,2α2Rfp2α2+Rf

), are given as follows:

∂2E[·]∂R2

D= − (p2α2+Rf )2

8(1−p2)α32R

3fγ

< 0

∂2E[·]∂WB∂`

=(2−p2)(qα1−p2α2)Rf+p2(p2α2−p1α1)Rf+p22(p1−q)α1α2

2(1−p2)Rf

∂2E[·]∂`∂RD

=(p2α2+Rf )2

4p2(1−p2)α32Rfγ

· ∂2E[·]∂WB∂`

+

+p2(p2α2+Rf )2·(p1−q)α1·[Rf−(2−p2)α2−2(1−p2)α2·γWBRf ]

8(1−p2)2α22R

2fγ

(A.2.11)

where the latter depiction has been chosen for convenience such that the following

relations can be immediately read off:

∂2E[WB(·,2)]∂WB∂`

∣∣∣∣`=0,RD=R∗D

< 0 ⇒ ∂2E[WB(·,2)]∂`∂RD

∣∣∣∣`=0,RD=R∗D

< 0

∂2E[WB(·,2)]∂`∂RD

∣∣∣∣`=0,RD=R∗D

> 0 ⇒ ∂2E[WB(·,2)]∂WB∂`

∣∣∣∣`=0,RD=R∗D

> 0(A.2.12)

since

Rf − (2− p2)︸︷︷︸>p2

α2

︸︷︷︸< 0 by (3.2.5) and (3.2.6)

− 2(1− p2)α2 · γWBRf < 0


whereas all other terms, except for∂2E[WB(·,2)]

∂WB∂ìtself, are always positive. But

Condition (3.3.2)

(2− p2)(qα1 − p2α2)Rf + p2(p2α2 − p1α1)Rf + p22(p1 − q)α1α2 ≤ 0

is equivalent to∂2E[·]∂WB∂`

< 0

such that, in line with (A.2.10), the unconstrained solution reacts to changes in WB

according tod`∗

dWB

< 0 anddR∗DdWB

> 0 .

However, since, obviously, the optimal loan-allocation rate leaves the unit interval

a change in WB implies that the optimal allocation rate actually remains at zero.

Anything else contradicts feasibility in the constrained problem according to (3.3.1).

Thus, the optimal value of R∗D does not change either such that we obtain

d`∗

dWB

= 0 anddR∗DdWB

= 0 .

By (3.2.48), the optimal total loan volume, L∗ changes in deed one-to-one for every

marginal change in WB.

Suppose now

The Constrained Problem

in the sense that the optimal loan-allocation rate of the unconstrained problem

is negative. Thus, Condition (3.3.3) strictly holds. Under Assumption (A.2.10),

an increase in WB meant a further decrease in the optimal loan-allocation rate of

the unconstrained problem. Even if the opposite holds and thus the optimal loan-

allocation rate locally increases, the allocation rate will remain negative and hence,

the optimal loan-allocation rate of the constrained problem, `∗, will be still kept

fixed at zero. As in either case ∂E[·]∂`

< 0 holds at ` = 0 (by having assumed that the

unconstrained optimum lie beneath zero), the implicit function theorem does not

apply and marginal changes of R∗D in WB can be either implicitly deduced from the

first-order Condition (A.2.8), or explicitly from the optimal Solution (A.2.9), both

clearly implying again


d`∗

dWB

= 0 anddR∗DdWB

= 0 .


Proof. As we have assumed Du(`, RD; 4) < WH in equilibrium, we have to consider

the following maximization problem for the bank under Case 41

max`,RD

E[WB(`, RD; 4)

]

s.t. Du(`, RD; 4) ·RD ≤ α1` [Du(`, RD; 4) +WB]

Du(`, RD; 4) ·RD ≤ α2(1− `) [Du(`, RD; 4) +WB]

implying the following first-order conditions

∂L[.]∂`

= − [q4RD−Rf+q4(1−q4)γWBR2D]·(p2α2−p1α1+α2λ2−α1λ1)

(1−q4)q4R2Dγ

!= 0

∂L[.]∂RD

=[α1`(p1+λ1)+α2(1−`)(p2+λ2)](2Rf−q4RD)−(q4+λ1+λ2)RfRD

(1−q4)q4R3Dγ

!= 0

λ1 · Du(`, RD; 4) ·RD − α1` [Du(`, RD; 4) +WB] != 0

λ2 · Du(`, RD; 4) ·RD − α2(1− `) [Du(`, RD; 4) +WB] != 0

Consider first λ1 = 0 and λ2 = 0; that is, none of the constraints constituting Case

4 according to (3.2.19) is binding. As the optimal interest rate will at least be equal

to the zero of the deposit-supply function, i.e. R∗D > 1q4Rf ≡ RD(4) (cf. (3.2.55))

given Case 42, we obtain∂L[.]

∂`≡ ∂E[.]

∂`< 0

due to the assumption p1α1 < p2α2. Thus, the bank chooses the lowest feasible

value for the loan-allocation rate ` given Case 4, implying

α1L∗1 = Du(`∗, R∗D; 4)R∗D .

This equality, in turn, contradicts the assumption that none of the constraints is

binding. As a consequence λ1 ≥ 0.

1We can neglect the short-sell conditions imposed on ` as the Conditions (3.2.19) for Case 4 arealways stricter.

2Note that the optimal interest rate strictly exceeds the deposit supply’s zero in RD inequilibrium (cf. Result 5). That is, RD = RD(4) is never an equilibrium.


Due to the constraints imposed by Definition (3.2.19) of Case 4, the bank chooses

its loan-allocation rate ` and RD such that

Du(R∗D) ·R∗D = α1L∗1 ≤ α2L

∗2

holds.

Let us proceed with α1L∗1 < α2L

∗2. Then we obtain λ2 = 0 and the first-order

conditions simplify to

∂L[.]

∂`= − [q4RD −Rf + q4(1− q4)γWBR

2D] · (p2α2 − p1α1 − α1λ1)

(1− q4)q4R2Dγ

!= 0

∂L[.]

∂RD

=[(p1 + λ1)α1`+ p2α2(1− `)] (2− q4RD)− (q4 + λ1)RfRD

(1− q4)q4R3Dγ

!= 0

Du(`, RD; 4) ·RD − α1` [Du(RD) +WB]!

= 0

Solving

Du(`, RD; 4) ·RD − α1` [Du(RD) +WB]!

= 0

for RD leads to two solutions in RD whereas exactly one solution is feasible:

R∗D(`) =q4α1`+Rf +

√(q4α1`+Rf )2 − 4q4Rfα1` [1− (1− q4)α1`γWB]

2q4 [1− (1− q4)α1`γWB].

Solving∂L[.]

∂RD

!= 0

for λ1 uniquely leads to

λ∗1(R∗D(`), `) =[p1α1`+ p2α2(1− `)] (2Rf − q4RD)− q4RDRf

RDRf − (2Rf − q4RD)α1`.

Thus, p2α2− p1α1−α1λ∗1(R∗D(`), `)

!= 0 is the necessary optimality condition for `

as the other factor of ∂L∂`

is always strictly positive. This equation is quadratic in `

and has as solutions

`(1) = 1(1−q4)α1γWB

`(2) =2p2α2Rf [(p2α2−Rf )(p1+p2−q)α1−(p2α2−p1α1)Rf ]

(p22α22−R2

f )q24α21−(p2α2−p1α1)2R2

f−2(p2α2−p1α1)q4α1R2f+4q4(1−q4)p22α

21α

22γWBRf

,

where the latter is the optimal loan-allocation rate and the former is unfeasible as

R∗D(`) diverges at ` = 1(1−q4)α1γWB

.


Plugging-in `∗ = `(2) into R∗D yields R∗D. Furthermore, the first Lagrange-Multiplier

simplifies to λ∗1 = p2α2−p1α1

α1emphasizing the cost to the risk-neutral bank to grant

the minimum feasible amount to Firm 1. The representation of `(2) emphasizes the

differences in expected gross returns and between expected returns to the risk-free

gross yield. For the representation in the result, (3.3.11), we re-arrange terms once

again.

Having the optimum (`∗, R∗D), we can use the relation α2(1− `∗) [Du(R∗D) +WB] ≥Du(R∗D)·R∗D to derive the lower bound that WB must exceed such that this inequality

strictly holds and thus the solution fulfills:

(p22α22−p21α2

1)R2f − q24α

21(p22α

22−R2

f ) − 2q4(p22α22−p1α1Rf )α1Rf + 4p22(1−q4)q4Rfα

21α

22γWB

4p22(1−q4)q4α21α2γRf

>

>p2α2(q4α1−Rf )−(p2−q)α1Rf

2p2q4(1−q4)α1α2γ.

As only the left-hand side of this inequality depends on WB and as it is strictly

increasing in WB, solving for WB results in (3.3.9). Is WB equal to or lower than

this bound, the inequality α2(1− `∗) [Du(R∗D) +WB] ≥ Du(R∗D) · R∗D becomes an

equality in equilibrium and thus the solution is characterized by

α1`∗ [Du(R∗D) +WB] = Du(R∗D) ·R∗D

and

α2(1− `∗) [Du(R∗D) +WB] = Du(R∗D) ·R∗D

resulting in

`∗ =α2

α1 + α2

and

R∗D =(α1+α2)Rf+q4α1α2+

√[(α1+α2)Rf−q4α1α2]

2+4q4(1−q4)α2

1α22γWBRf

2q4[α1+α2−(1−q4)α1α2γWB ].

By `∗ = α2

α1+α2and λ∗1 = p2α2−p1α1

α1+ α2

α1λ∗2 from ∂L[.]

∂`

!= 0 , the remaining first-order

condition, ∂L[.]∂RD

!= 0, becomes

α21α2(p1 + p2)(2Rf − q4R

∗D)− q4α1(α1 + α2)R∗DRf −

− (p2α2 − p1α1) [(α1 + α2)R∗DRf − α1α2(2Rf − q4R∗D)]

!= 0 ,

yielding

λ∗2 =p2α1α2 − [(p2 − q)α1 + p2α2]R∗DRf − p2α1α2(q4R

∗D −Rf )

α1α2(q4R∗D −Rf ) + [(α1 + α2)R∗D − α1α2]Rf

.


Finally, we derive the bounds on `∗ and on R∗D. Let us show why

`(2) ∈ (0,α2

α1 + α2

) ,

holds, i.e. why the bounds on the loan-allocation rate hold as given in (3.3.11). `(2)

strictly decreases in WB with

limWB→∞

`(2) = 0+ .

Second, `(2) becomes

`(2) =α2

α1 + α2

,

at the point where WB is equal to the Bound (3.3.9). Thus, the upper bound on `(2)

is established.

RD from (3.3.10) strictly increases in WB. For the border case WB = 0, R∗D =1q4Rf > Rf , implying D∗ = 0 as RD(4) = 1

q4Rf . Assumption (3.2.7) is crucial here

to obtain R∗D = 1q4

as it ensures (α1 +α2)Rf−q4α1α2 > 0 in connection with Rf ≥ 1.

If (α1 + α2)Rf − q4α1α2 were negative, R∗D would become α1α2

α1+α2at WB = 0.

If WB equals the border given by (3.3.9), R∗D according to (3.3.10) becomes identical

to the formula given in (3.3.11). Thus, the latter also exceeds RD(4).


Proof. If Condition (3.3.9) is satisfied both inequalities defining Case 4 according to

(3.2.19) hold with equality. In other words, an optimal and feasible solution under

Case 4 is effectively constrained by WB. Let us consider the following first-order

conditions, starting with the partial derivative of the Lagrangian with respect to

RD:

− q4 [Du(RD) +WB] + [p1α1`+ p2α2(1− `)− q4RD] · ∂Du(RD)∂RD

− . . .− λ1

[(RD − α1`)

∂Du

∂RD+Du(RD)

]− λ2

[(RD − α2(1− `)) ∂Du

∂RD+Du(RD)

]!

= 0

λ1 · [Du(RD)RD − α1`(Du(RD) +WB)]

!= 0

λ2 · [Du(RD)RD − α2(1− `)(Du(RD) +WB)]!

= 0


As the bounds to Case 4 are binding, we obtain R∗D ≥ α1`∗ and R∗D ≥ α2(1 − `∗)

where equality holds if and only if WB = 0. Furthermore, calculations show

(RD − α1`)∂Du

∂RD+Du(RD) =

[R∗D−α1`∗]+α1`∗(q4R∗D−Rf )

γq4(1−q4)(R∗D)3> 0

[RD − α2(1− `∗)] ∂Du∂RD

+Du(RD) =[R∗D−α2(1−`∗]+α2(1−`∗)(q4R∗D−Rf )

γq4(1−q4)(R∗D)3> 0

The positive signs are due to the binding conditions induced by (3.2.19) and by the

fact that R∗D > 1q4Rf ≡ RD(4) holds as otherwise we did not consider an optimal

outcome (cf. Result 5). As λ1 ≥ 0 and λ2 ≥ 0 must hold due to the binding

constraints, too, we obtain

−λ1

[(RD − α1`)

∂Du

∂RD

+Du(RD)

]− λ2

[(RD − α2(1− `)) ∂D

u

∂RD

+Du(RD)

]≤ 0 .

Hence,

−q4 [Du(RD) +WB] + [p1α1`+ p2α2(1− `)− q4RD] · ∂Du(RD)

∂RD

≥ 0

holds, and in particular∂Du(RD)

∂RD

> 0 . (A.2.13)

With respect to WB, the optimal deposit interest rate R∗D is of the form f(WB)

where

f(WB) =a+√b+ c ·WB

d− e ·WB

with a, b, c, d, and e strictly positive constants appropriately defined. Obviously,

f(WB) (R∗D, respectively) strictly increases in WB. Altogether, we obtain

dDu(R∗D)

dWB

=∂Du(RD)

∂RD

· ∂R∗D

∂WB

> 0 . (A.2.14)



Proof. As we have assumed Du(`, RD; 4) < WH in equilibrium, the maximization

problem under Case 4 if both firms’ projects are equal is given by

max`,RD

E[WB(`, RD; 4)

]

s.t. Du(`, RD; 4) ·RD ≤ α` [Du(`, RD; 4) +WB]

Du(`, RD; 4) ·RD ≤ α(1− `) [Du(`, RD; 4) +WB]


∂L∂`

= − [(2p−q)RD−Rf+(2p−q)(1−2p+q)γWBR2D]·α·(λ2−λ1)

(2p−q)(1−2p+q)R2Dγ

!= 0

∂L∂RD

=[p+`λ1+(1−`)λ2][2Rf−(2p−q)RD]α−[(2p−q)+λ1+λ2]RfRD

(2p−q)(1−2p+q)R3Dγ

!= 0

λ1 · Du(`, RD; 4) ·RD − α` [Du(`, RD; 4) +WB] != 0

λ2 · Du(`, RD; 4) ·RD − α(1− `) [Du(`, RD; 4) +WB] != 0

By the first condition we obtain λ∗1 = λ∗2 reflecting the equality of costs imposed by

Case 4 if both projects are equal. Hence,

∂L∂`

= 0

holds for all optimal choice under Case 4 regardless of the structures of the optimal

loan-allocation rate and the optimal deposit interest rate.

Consider now λ1 = 0 and λ2 = 0; that is, none of the constraints constituting Case

4 according to (3.2.19) is binding. Then we obtain

∂L∂RD

=p [2Rf − (2p− q)RD]α− (2p− q)RfRD

(2p− q)(1− 2p+ q)R3Dγ

!= 0

resulting in

R∗D =2pαRf

(2p− q)(pα +Rf ).

Plugging-in R∗D into the Case constraints yields the unique lower and the unique

upper bound on the optimal loan-allocation rate `∗, respectively, given in (3.3.15).

Analogous to the proof to Result 9 given in Appendix A.2.1.5 we obtain the threshold

value for WB (3.3.12) above which both Case constraints never hold with equality

in equilibrium.


Consequently, if WB is equal or below this threshold value, both Case constraints

must hold with equality in equilibrium and by the projects’ equality we obtain

`∗ = 12. As both Case constraints are binding, the optimal interest rate R∗D solves

α 12

[Du(R∗D +WB]!

= Du(R∗D) ·R∗D.

The Lagrange multipliers λ∗1 = λ∗2 = λ∗ can be obtained from

(p+ λ∗) [2Rf − (2p− q)R∗D]α− [(2p− q) + 2λ∗]RfR∗D

!= 0

after having simplified the numerator of ∂L[.]∂RD

, yielding

λ∗ =[pα− (2p− q)R∗D]Rf − [(2p− q)R∗D −Rf ] pα

[(2p− q)R∗D −Rf ]α + (2R∗D − α)Rf

.

Finally, let us analyze the bounds of `∗ and R∗D. The optimal deposit interests rates

are special cases of those shown by Result 9. Thus the bounds and the behavior in

WB are given in an analogous way.

Concerning `, let us consider the lower bound first,

2pRf (pα−Rf )

q4(p2α2 −R2f ) + 4p2α2(1− q4)γWBRf

.

If WB equals the bound stated in (3.3.12), this term becomes 12. As it strictly

decreases in WB, it is bounded from below by zero, i.e. by its limit value for WB →∞. Even if WB = 0 were allowed for this solution, this bound would remain lower

than one:

2pRf (pα−Rf )

q4(p2α2 −R2f )

=2pRf

q4(pα +Rf )< 1

q4≡2p−q⇔ qRf < (2p− q)pα .

The upper bound stated in (3.3.15),

(pα−Rf ) [q4pα− qRf ] + 4p2α2(1− q4)γWBRf

q4(p2α2 −R2f ) + 4p2α2(1− q4)γWBRf

becomes 12

if WB is equal to the bound stated in (3.3.12), too. It strictly increases

in WB and is bounded from above by its limit value for WB →∞. This limit value

is equal to one. Taking the derivative of this upper bound w/r/t WB shows that the

strict monotonicity in WB is guaranteed by

q4(p2α2−R2f )−(pα−Rf ) [q4pα− qRf ] = (pα−Rf )(q4+q)Rf = 2p(pα−Rf )Rf > 0 .


Consider Equilibrium (3.3.13). After some algebraic manipulation, the sensitivity

of R∗D, given by (3.3.14), w/r/t q4 can be displayed as follows,

q4αγWB

√Q∗4(p,q,α)+2Rf [2−(1−q4)αγWB ]

q4[2−(1−q4)αγWB ]√Q∗4(p,q,α)

·R∗D +(2−αγWB)αRf

q4[2−(1−q4)αγWB ]√Q∗4(p,q,α) ,

where Q∗4(p, q, α) = [2Rf − q4α]2 + 4q4(1− q4)α2γWBRf . This expression for∂R∗D∂q4

is positive since 2Rf ≥ αRf due to (3.2.7) and since 2− (1−q4)αγWB > 2−αγWB

due to q4 > 0 hold.

Consider Equilibrium (3.3.15). The sensitivities of R∗D w/r/t q, p, and α can be

expressed as follows:

∂R∗D∂q

=∂R∗D∂q4︸︷︷︸<0

· ∂q4

∂q︸︷︷︸=−1

> 0 ,

∂R∗D∂p

=−2αRf (2p

2α + qRf )

q24(pα +Rf )2

< 0 , and

∂R∗D∂α

=2pR2

f

q4(pα +Rf )2> 0 .


Proof. Suppose that the household is not constrained by its initial wealth WH . The

sign of the expression qjRD(RD − µj) − σ2j determines the sign of the derivative

∂Du(·)∂WB

.

Let us consider Case 1 first. Since both projects are assumed to be equal, i.e. p1 =

p2 = p and α1 = α2 = α, the expression q1RD(RD − µ1)− σ21 becomes

(p− q)α` [µ1 − α`] + (p− q)α(1− `) [µ1 − α(1− `)] .

As the optimal loan-allocation rate `∗ equals 12, this expression further simplifies to

(p− q)α ·[µ1 − α

1

2

].

Due to `∗ = 12

according to Result 6 and αi ≤ 2 according to (3.2.7), the relation

α`∗ ≤ 1 holds. Suppose now that µ1 < α`∗ ≤ 1 hold. As a consequence, we

obtain µ1 < Rf as Rf ≥ 1. Thus, Du(`∗, R∗D) < 0 and by excluding short-selling


Ds(`∗, R∗D) = 0 contradicting Ds(`∗, R∗D) > 0 stated by Result 5. Thus µ1 > α`∗ ≥ 1

holds in (12, R∗D, 1) and hence

∂Ds(·)∂WB

> 0 (A.2.15)

in equilibrium.

Concerning Case 2 and 3 and `∗ 6= 0 or, `∗ 6= 1, Result 4 applies. Only if the bank

chooses a corner solution in the sense that either `∗ = 0 (Case 2), or `∗ = 1 (Case

3) holds, the dependence of the deposit volume from bank capital vanishes and thus

the cushion effect vanishes, too. In essence, the same argumentation holds true if

Case 4 prevails. These notions have been already captured under Result 1.

A.2.2 The Equilibrium with Regulation by Fixed Risk

Weights


Proof. The first-order conditions to the maximization problem considered in Result

14 are:

q · (α−RD) · ∂Du(`, RD; 1)

∂`!

= λ ·[∂Du(`, RD; 1)

∂`− k′(`)WB

]

q · (α−RD) · ∂Du(`, RD; 1)

∂RD

− q ·Du(`, RD; 1)!

= λ · ∂Du(`, RD; 1)

∂RD

λ · [Du(`, RD; 1) − k(`) ·WB]!

= 0

As both loans exhibit equal characteristics, especially concerning their credit risk, we

have c1 = c2 and hence k(`) = 1−cc1cc1

, i.e. k(`) is independent of the loan-allocation

rate `. Let us rewrite the first-order conditions as

[q · (α−RD)− λ] · ∂Du(`, RD; 1)

∂`!

= 0

[q · (α−RD)− λ] · ∂Du(`, RD; 1)

∂RD

!= q ·Du(`, RD; 1)

λ · [Du(`, RD; 1) − k(`) ·WB]!

= 0

Let us discuss all possible cases that are able to solve this non-linear system of

equation.

First, let us consider q·(α−RSD)−λS = 0. This implies coercively Du(`S, RS

D, 1) = 0

by the second equation. The third equation thus necessitates λS = 0 resulting


recursively in RSD = α. Then the bank’s expected final wealth is only qαWB as it

forgoes any expected gains from intermediation although the regulatory constraint

has not been (fully) exploited. Thus, a solution such as this contradicts optimality.

Second, we consider q · (α−RSD)−λS 6= 0. Thus we obtain ∂Du(`,RD;1)

∂`= 0 uniquely

resulting in `S = 12. Furthermore, q · (α−RS

D)− λS 6= 0 excludes Du(`, RD; 1) = 0

as the bank would forgo expected profits from intermediation without being forced

to do so. By the short-sell constraint on deposits we obtain Du(12, RS

D, 1) > 0 which

implies two subcases to consider.

First, we assume q · (α−RSD)−λS < 0 implying ∂Du(`,RD;1)

∂RD< 0. By the first-order

condition with respect to RD we obtain

∂E[WB(·)

]

∂RD

= q · (α−RD) · ∂Du(1

2, RD; 1)

∂RD

− q ·Du(1

2, RD; 1) < 0

because of α − RD > 0 as anything else would contradict optimality. By Result 5,

the optimal interest rate R∗D is unique for given loan-allocation rates. Here, we have

just shown the first derivative of the unregulated bank’s objective function. As this

derivative is negative for (12, RS

D), uniqueness, as shown in the proof A.2.1.1, implies

that RSD > R∗D for ` = 1

2. Because of the last first-order condition given by the

regulatory constraint,

Ds(1

2, RS

D, 1) ≤ 1− cc1

cc1

·WB

must hold. As we have assumed that regulation is binding, we have Ds(12, RS

D, 1) <

D∗. Thus, RSD must have already surpassed the critical rates RD(1), and R

u

D(1),

respectively, where the latter is defined by (3.2.52). However, a deposit interest rate

with RSD > RD(1) > R∗D contradicts optimality as the bank could choose a rate R′D,

RD(1) > R∗D > R′D, such thatDs(12, R′D, 1) = 1−cc1

cc1·WB holds, i.e. such that the same

deposit volume as before can be attained, but that leads to a wider spread α−RD

that the bank can earn through intermediation. Note that the existence of such an

interest rate R′D is due to the hump-shaped deposit-supply function characterized

by Result 2. Thus, the subcase just considered contradicts optimality as well.

The second subcase and the only remaining case to consider is given by q · (α −RSD)− λS > 0. In analogy to the subcase just discussed we are now obtaining

∂Du(12, RD; 1)

∂RD

> 0

Consequently, a deposit interest rate RSD with R∗D < RS

D < RD(1) contradicts


the assumption of a binding regulation. Thus, given∂Du( 1

2,RD;1)

∂RD> 0 only

an interest rate RSD with R∗D > RS

D may satisfy the regulatory constraint λ ·[Du(1

2, RD; 1) − 1−cc1

cc1·WB

]= 0. Furthermore, RS

D > RD(1) holds as otherwise

the bank forgo expected from profits from intermediation without being forced to

do so.

As RD(1) < RSD < R∗D,

q · (α−RSD) · ∂D

u(12, RS

D, 1)

∂RSD

− q ·Du(1

2, RS

D, 1)

holds, we obtain λS > 0 and thus DS = 1−cc1cc1·WB.

Note that the objective

E[WB(`, RD; 1)

]= q · (α−RD) ·Du(`, RD; 1) + qαWB

cannot be heightened if corner solutions for RD, such as RD(1) or α, would be chosen

for any fixed loan-allocation rate ` that is compatible with Case 1. Furthermore,

the following partial derivative becomes by the projects’ equality

∂Du(`, RD; 1)

∂`=

∂Du(`, RD; 1)

∂σ2· ∂σ

2

∂`

implies `S = 12

as solution to the appropriate first-order condition of the bank for

any given RD. Furthermore, the second partial derivative of the bank’s objective

function with respect to ` becomes

∂2E[WB(`,RD;1)]∂`2

∣∣∣`= 1

2

= q · (α−RD) ·[∂2Du(·)∂(σ2

1)2·(∂σ2

1

∂`

)2

+ ∂Du(·)∂σ2

1· ∂2σ2

1

∂`2

∣∣∣∣`= 1

2

]

= q · (α−RD) · ∂Du(·)∂σ2

1· ∂2σ2

1

∂`2< 0

confirming that `S = 12

is a local maximizer given any RD. As `S = 12

is the only

critical point, it is also the only global maximizer with respect to the loan-allocation

rate for any deposit interest rate RD. Thus, we obtain as solution `S = 12

and

RSD =

cc1 + (2− cc1)(p− q)αγWB −√QS

1 (c1, α)

2(1− cc1)(1− q)γWB

where QS1 (c1, α) is as given by the result. The sensitivities of RS

D are derived by total

differentiation of the optimal deposit volume. By `S = 12, we obtain LS = 1

cc1·WB


and hence DS = 1−cc1cc1·WB. Since

dDS

dθ≡ ∂Ds(1

2, RS

D; 1)

∂θ+

∂Ds(12, RS

D; 1)

∂RD︸︷︷︸>0 by Res.(5)

·∂RSD

∂θ

holds for any parameter θ, we subsequently obtain the signs for the deposit interest

rate’s sensitivities, as shown in the result. Specifically, for θ = γ,

∂Ds(12, RS

D; 1)

∂γ< 0

holds, resulting in∂RS

D

∂γ> 0

because ofdDS

dγ= 0 .

Since RSD is a function of γ ·WB, and γ,WB > 0,

∂RSD

∂WB

> 0

must be true, too. Analogous to γ, we can proceed with Rf . Finally,

dDS

dc= − 1

c2c1

·WB < 0

results in connection with∂Ds(1

2, RS

D; 1)

∂c= 0

in∂RS

D

∂c< 0 .

The same reasoning applies to c1 and c2, respectively, as they are assumed to be

identical.


Proof. By assumption, `S = 0 and hence jS = 2 is optimal. Thus,

∂E[WB(0, RD; 2)]

∂RD

= λ · ∂Ds(0, RD; 2)

∂RD

λ · [Ds(0, RD; 2)− k(0) ·WB] = 0


must be solved for λ and RD yielding

λS =p2 ·

[α2 ·

√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf − cc2Rf

]

cc2 ·√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

RSD =

cc2p2 −√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

2(1− cc2)p2(1− p2)γWB

.

The sensitivity of RSD with respect to WB is thus

∂RSD

∂WB

=1

WB

·(

cc2√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

·Rf − RSD

).

∂RSD∂WB

is thus positive if

cc2√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

·Rf > RSD

holds. Plugging-in the formula for RSD and multiplying this inequality cross-wise,

i.e. the left-hand side with the denominator of RSD and the right-hand side by the

square-root term yields:

2cc2(1− cc2)p2(1− p2)γWBRf

>

cc2p2 ·√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf −

− c2c22p

22 + 4cc2(1− cc2)p2(1− p2)γWBRf .

Simplifying and re-arranging terms yields

cc2p2 −√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf > 2(1− cc2)(1− p2)γWBRf ,

which is true if and only if

RSD > Rf

holds.

Let us show that even RSD > 1

p2·Rf holds which is the case if and only if

cc2p2−√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf > 2(1−cc2)p2(1−p2)γWB ·

1

p2

Rf


is true. Note that cc2 ≤ 0.08 · 1.5 = 0.12 < 1. Re-ordering yields

cc2p2 − 2(1− cc2)(1− p2)γWBRf >√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf .

Assume that the term under the square-root is positive (as a result, the left-hand

is then positive, too, which can be easily seen if the left-hand side is multiplied by

cc2p2 > 0). Taking the square of both sides leads to

c2c22p

22 + 4(1− cc2)2(1− p2)2γ2W 2

BR2f − 4cc2(1− cc2)p2(1− p2)γWBRf >

> c2c22p

22 − 4cc2(1− cc2)p2(1− p2)γWBRf

⇔4(1− cc2)2(1− p2)2γ2W 2

BR2f > 0 ,

which is always true.

Finally, let us show that the term under the square-root must be positive if regulation

is binding in the case here considered. The term under the square root strictly

decreases in WB. Thus, the highest value WB can take such that the whole square

root remains in the reals, is given by

c2c22p

22

!= 4cc2(1− cc2)p2(1− p2)γWBRf .

Plugging-in into RSD yields

RSD =

2Rf

p2

, if WB =cc2p2

4(1− cc2)(1− p2)γRf

,

resulting in

DS =p2

4(1− p2)γRf

.

But this deposit volume is equal to

DS =1− cc2

cc2

·WB

iff

cc2p2 = 4(1− cc2)(1− p2)γWBRf

holds, i.e. iff the assumption

c2c22p

22

!= 4cc2(1− cc2)p2(1− p2)γWBRf


is fulfilled. In total,∂RS

D

∂WB

> 0

has been thus shown.

Combining∂RSD∂WB

> 0 with the sensitivity of the optimal deposit volume with respect

to RSD,

∂Ds(0, RSD, 2)

∂RSD

=2Rf − p2R

SD

γp2(1− p2)(RSD)3

results in the sensitivity of the total loan volume with respect to WB. The other

sensitivities of RSD are as follows:

∂RSD

∂α2

= 0

∂RSD

∂p2

= − cc2 ·[p2R

SD −Rf

1− p2

+1

p2

Rf

]

∂RSD

∂Rf

=cc2√

c2c22p

22 − 4cc2(1− cc2)p2(1− p2)γWBRf

∂RSD

∂c2

= − c · (p2RSD −Rf )

(1− cc2) ·√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)γWBRf

Alternatively, the signs of the sensitivities of the deposit interest rate RSD can be

determined according to

∂Ds(0, RSD, 2)

∂RSD

· ∂RSD

∂θ+

∂Ds(0, RSD, 2)

∂θ=

∂(

1cc2WB

)

∂θ,

is regulation is assumed to be binding and `S to be fixed to zero.


Proof. According to Result 8, the sensitivity of the deposit interest rate with respect

to the risk-free rate is given by

∂R∗D∂Rf

=2p2α

22

(p2α2 +Rf )2

and according to Result 16 that under regulation by

∂RSD

∂Rf

=cc2√

c2c22p

22 − 4cc2(1− cc2)p2(1− p2)RfγWB


Suppose now that∂R∗D∂Rf

>∂RS

D

∂Rf

holds. That is equivalent to

2p2α22 ·√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)RfγWB > cc2 · (p2α2 +Rf )

2

⇔p2α

22 ·√. . .− cc2R

2f − 2cc2p2α2Rf > p2α

22

(cc2p2 −√. . .

)

⇔

−p2α2

2Rf︸︷︷︸≥2

cc2 − α2︸︷︷︸≤2

·√. . .

︸︷︷︸≥2·[cc2−

√...]>0

−cc2R2f > p2α

22

(cc2p2 −√. . .

)︸︷︷︸

>0

,

resulting in a contradiction as especially

cc2p2 −√c2c2

2p22 − 4cc2(1− cc2)p2(1− p2)RfγWB > 0

holds since RSD > 0 is true. Thus, we obtain

0 <∂R∗D∂Rf

<∂RS

D

∂Rf

,

meaning that the deposit interest rate becomes pro-cyclical by regulation concerning

changes in the risk-free rate according to Definition (2.2.1).


Proof. Case 4 requires

α1`S(DS +WB) ≥ DSRS

D and α2(1− `S)(DS +WB) ≥ DSRSD

to hold. As regulation is supposed to be binding and by having assumedDS < WH to

hold in equilibrium, the deposit volume satisfies (3.3.21) and both Case constraints

become

α1`S ≥

1− c ·

[c1`

S + c2(1− `S)]·RS

D

and

α2(1− `S) ≥

1− c ·[c1`

S + c2(1− `S)]·RS

D


resulting for RSD in

RSD ≤ min

α1`

S

1− c · [c1`S + c2(1− `S)],

α2(1− `S)

1− c · [c1`S + c2(1− `S)]

.

Due to α2 ≥ α1, RSD can be at most

RSD ≤ min

`S, (1− `S)

· α2

1− c · [c1`S + c2(1− `S)].

Because of ` ∈ [0, 1], the highest upper bound to min`S, (1− `S) is a half, yielding

RSD ≤

α2 · 12

1− c · [c1`S + c2(1− `S)].

This upper bound may not go below the zero of the deposit-supply function in Case

4, i.e. not below 1q4Rf . If it did, DS = 0, being a contradiction to (3.3.21) with

WB > 0. Hence, α2 must exceed the threshold

α2 >2 ·

1− c ·[c1`

S + c2(1− `S)]

q4

·Rf

c=0.08,ci≤1.5

≥ 1.76

q4

·Rf

if regulation shall be binding in Case 4.



the following maximization problem for the bank under Case 4 under regulation

with fixed risk weights

max`,RD

E[WB(`, RD; 4)

]

s.t. Du(`, RD; 4) ·RD ≤ α1` [Du(`, RD; 4) +WB]

Du(`, RD; 4) ·RD ≤ α2(1− `) [Du(`, RD; 4) +WB]

c · [c1`+ c2(1− `)] [Du(`, RD; 4) +WB] ≤ WB



∂L∂`

= − [q4RD−Rf+q4(1−q4)γWBR2D]

(1−q4)q4R2Dγ

·· [α2(p2 + λ2)− α1(p1 + λ1) + c(c1 − c2)λ3]

!= 0

∂L∂RD

= − (q4+λ1+λ2)RfRD+q4RD·λ3c[c1`+c2(1−`)](1−q4)q4R3

Dγ

+α1`(p1+λ1)+α2(1−`)(p2+λ2)−λ3c[c1`+c2(1−`)](2Rf−q4RD)

(1−q4)q4R3Dγ

!= 0

λ1 · Du(`, RD; 4) ·RD − α1` [Du(`, RD; 4) +WB] != 0

λ2 · Du(`, RD; 4) ·RD − α2(1− `) [Du(`, RD; 4) +WB] != 0

λ3 · c · [c1`+ c2(1− `)] [Du(`, RD; 4) +WB]−WB != 0

By assumption, regulation is binding. As both projects are different in expected

returns and in variance, at least one Case constraint is binding, cf. Result 9 and

Appendix A.2.1.5.

Let (3.3.33) prevail. Then, solving

c · [c1`+ c2(1− `)] [Du(RD) +WB]!

= WB

α1` [Du(RD) +WB]!

= Du(RD) ·RD

for ` and RD leads to (3.3.34) after having excluded unfeasible outcomes. The

shadow cost of regulation amounts to

λS3 =p2α2

[(α1 −RS

D)Rf − (q4RSD −Rf )α1

]− (p2 − q)α1R

SDRf

c [c2α1(2Rf − q4RSD)− (c2 − c1)RS

DRf ]≥ 0

whose positive sign results from the assumption that regulation is strictly binding.

The numerator is positive if and only if

RSD ≤

2p2α1α2Rf

(p2α2 − p1α1)Rf + q4α1(p2α2 +Rf )

(3.3.11)≡ R∗D

is satisfied, where the latter equivalence refers to the optimal choice in Case 4 without

regulation that results in α1L∗1 < α2L

∗2. The denominator is strictly positive if and

only if

RSD <

2c2α1Rf

(c2 − c1)Rf + c2q4α1

is satisfied which is the case for RSD < R∗D as R∗D <

2c2α1Rf(c2−c1)Rf+c2q4α1

holds. This

inequality can be traced back to 0 < c1p2α2 +c2(p2−q)α1. Thus, binding regulation


implies a reduction in the deposit rate.3

The shadow cost of granting a minimum to Firm 1 equals

λS1 =(c1p2α2 − c2p1α1)(2Rf − q4R

SD) + (c2 − c1)q4R

SDRf

c2α1(2Rf − q4RSD)− (c2 − c1)RS

DRf

.

Hence, c1p2α2−c2p1α1 ≥ 0 is sufficient to make the constraint α1` [Du(RD) +WB] ≥Du(RD) ·RD be binding as c2 > c1 has been assumed. If c2 = c1, c1p2α2−c2p1α1 ≥ 0

is necessary, and solving the first-order conditions results in (3.3.36) for `S and

(3.3.34) again for RSD. Note that (3.3.34) is not feasible for `S if c1 = c2 holds.

Likewise, if (3.3.38) prevails, the Solution (3.3.39) can be obtained and the associated

Lagrange multipliers are given by

λS3 =p1α1

[(α2 −RS

D)Rf − (q4RSD −Rf )α2

]− (p1 − q)α2R

SDRf

c [c1α2(2Rf − q4RSD) + (c2 − c1)RS

DRf ]≥ 0 ,

and

λS2 =(c2p1α1 − c1p2α2)(2Rf − q4R

SD)− (c2 − c1)q4R

SDRf

c1α2(2Rf − q4RSD) + (c2 − c1)RS

DRf

which is positive if

c2p1α1 − c1p2α2 >(c2 − c1)q4R

SDRf

2Rf − q4RSD

holds. Lacking un upper bound to RSD that is tight enough, we simply plug-in RS

D

into the formula from above to obtain the following condition for λS2 > 0 to hold:

p1α1

p2α2≥ c1

c2+ c2−c1

c2· Rfp1α1· c(c2−c1)Rf−cc1q4α2+

√QS4 (c2,c1,α2)

3cc1(c2−c1)Rf+cc1q4α2−4(1−cc1)(1−q4)α2γWBRf−√QS4 (c2,c1,α2)

where

QS4 (c2, c1, α2) = c2 [c1q4α2 + (c2 − c1)Rf ]

2 − 4cc1(1− cc1)(1− q4)q4α22γWBRf .

3Instead of arguing for RSD < R∗D via the Lagrange-multiplier associated with the regulatory

constraint we can make use of the fact that DS < D∗ holds by the binding regulation and of thefact that the deposit-supply function depends on RD only under Case 4. Furthermore, we mustconsider that the slope of Du(RD) is positive under both regimes (cf. Result 5, Result 2, andRemark 1.



Proof. Let us derive the sign of the sensitivity of RSD with respect to WB. If

Condition (3.3.37) holds, RSD is given by (3.3.39) and is thus a function of WB

of the following form:

RSD =

a+√b− c ·WB

d− e ·WB

with a, b, c, d, e > 0 appropriately defined. Then we obtain

∂RSD

∂WB

=

12· 1√

b−c·WB· (−c) · (d− ex) +

(a+√b− c ·WB

)· e

(d− e ·WB)2=

=ae+ e

√b− c ·WB − c

2√b−c·WB

· (d− ex)

(d− e ·WB)2=

=2ae√b− c ·WB + 2e(b− c ·WB)− c · (d− ex)

2√b− c ·WB(d− e ·WB)2

=

=2ae√b− c ·WB + 2eb− ce ·WB − cd2√b− c ·WB(d− e ·WB)2

=

=2ae√b− c ·WB + e(

>0︷︸︸︷b− c ·WB) + be− cd

2√b− c ·WB(d− e ·WB)2

.

This derivative is positive if be > cd holds. The expression be− cd can be simplified

to yield

2c2(1− cc1)(1− q4)q4α2γWB ··[c2

1q24α

22 + (c2 − c1)2R2

f + 2c1(c2 − c1)q4α2Rf − 4c1(c2 − c1)α2q4

],

of which the first factor is strictly positive, while the second factor is bounded from

below by

[c2

1q24α

22 + (c2 − c1)2R2

f + 2c1(c2 − c1)q4α2Rf − 4c1(c2 − c1)q4α2

]≥

≥ [c21q

24α

22 + (c2 − c1)2 + 2c1(c2 − c1)q4α2 − 4c1(c2 − c1)q4α2] =

= [c1q4α2 − (c2 − c1)]2 ≥ 0 ,

due to Rf ≥ 1 according to (3.2.6) and due to c2 > c1 by Assumption (3.3.32). As

a consequence, we obtain∂RSD∂WB

> 0.




the following maximization problem for the bank under Case 4 under regulation

with fixed risk weights if both firms’ projects are equal:

max`,RD

E[WB(`, RD; 4)

]

s.t. Du(`, RD; 4) ·RD ≤ α` [Du(`, RD; 4) +WB]

Du(`, RD; 4) ·RD ≤ α(1− `) [Du(`, RD; 4) +WB]

c · c1 [Du(`, RD; 4) +WB] ≤ WB


∂L∂`

= − [(2p−q)RD−Rf+(2p−q)(1−2p+q)γWBR2D]·α·(λ2−λ1)

(1−2p+q)(2p−q)R2Dγ

!= 0

∂L∂RD

= − [(2p−q)−(λ1+λ2)]RDRf+cc1[2Rf−(2p−q)RD]λ3

(1−q4)q4R3Dγ

+[2Rf−(2p−q)RD][p−`λ1−(1−`)λ2]α

(1−q4)q4R3Dγ

!= 0

λ1 · Du(`, RD; 4) ·RD − α` [Du(`, RD; 4) +WB] != 0

λ2 · Du(`, RD; 4) ·RD − α(1− `) [Du(`, RD; 4) +WB] != 0

λ3 · cc1 [Du(`, RD; 4) +WB]−WB != 0

Regulation is binding by assumption. Suppose none of the Case constraints is

binding, i.e.

αLS1 > DSRSD and αLS2 > DSRS

D .

implying λS1 = λS2 = 0, and the remaining first-order conditions become:

∂L∂RD

=[2Rf−(2p−q)RD]pα−(2p−q)RDRf−cc1[2Rf−(2p−q)RD]λ3

(1−q4)q4R3Dγ

!= 0

λ3 · cc1 [Du(`, RD; 4) +WB]−WB != 0

Solving the regulatory constraint for RD leads to two solutions whereas one is

the optimal one as given in Result 21. The Lagrange multiplier of the regulatory

constraint becomes

λS3 =2pαRf − (2p− q)RS

D(pα +Rf )

cc1 [2Rf − (2p− q)RSD]

.


The denominator is positive since 2Rf − (2p − q)RSD > 2Rf − (2p − q)α ≥ 2Rf −

(2p− q)2 ≥ 2 · 1− 1 · 2 = 0 holds. The numerator is positive if and only if

RSD <

2pαRf

(2p− q)(pα +Rf )

holds whereas the right-hand side is the promised deposit interest rate without

regulation with the Case constraints not binding. Hence, by Result 20, we obtain

λS3 > 0 and RSD < R∗D.

Finally, the constraints given by the result yield the lower and upper bound for all

loan-allocation rates that are feasible under this equilibrium.

Again, the sensitivities of RSD can be derived by total differentiation of the optimal

deposit volume. By `S = 12, we obtain LS = 1

cc1·WB and hence DS = 1−cc1

cc1·WB.

SincedDS

dθ≡ ∂Ds(RS

D; 4)

∂θ+

∂Ds(RSD; 4)

∂RD︸︷︷︸>0 by Res.(5)

·∂RSD

∂θ

holds for any parameter θ, we subsequently obtain the signs for the deposit interest

rate’s sensitivities, as shown in the result. Specifically, for θ = γ,

∂Ds(RSD; 4)

∂γ< 0

holds, resulting in∂RS

D

∂γ> 0

because ofdDS

dγ= 0 .

Since RSD is a function of γ ·WB, and γ,WB > 0,

∂RSD

∂WB

> 0

must be true, too. Equivalently, we can argue by

dDS

dWB︸︷︷︸>0

≡ ∂Ds(RSD; 4)

∂WB︸︷︷︸≡0

+∂Ds(RS

D; 4)

∂RD︸︷︷︸>0

· ∂RSD

∂WB

in Case 4, cf. Res. 1.

Analogous to γ, we can proceed with Rf . Concerning the regulatory parameters c


and ci, we refer to Appendix A.2.2.1.

A.2.3 The Equilibrium with Regulation by a Value-at-Risk

Approach

A.2.3.1 The Regulatory Constraints

As L = α1X1L1 + α2X2L2 holds, the VaR constraint is in detail given by

P[

E[α1X1L1 + α2X2L2]− (α1X1L1 + α2X2L2) ≥ τWB

]≤ p ,

with E(α1X1L1 + α2X2L2) = p1α1L1 + p2α2L2. To derive conditions that assure

that a given level of confidence p is fulfilled, let us rearrange the probability to

P[L ≤ p1α1L1 + p2α2L2 − τWB

]≤ p .

Thus, the bank complies with a fixed p if L does not surpass the threshold p1α1L1 +

p2α2L2 − τWB too often. This, in turn, depends on the relative magnitude of both

loan redemptions to each other.

Let us first consider the case α1L1 < α2L2. By the joint distribution of (X1, X2),

we obtain the cumulative distribution function of the random variable L as

P[0 ≤ L < α1L1] = 1− p1 − p2 + q

P[L ≤ α1L1] = P[L < α2L2] = 1− p2

P[L ≤ α2L2] = P[L < α1L1 + α2L2] = 1− qP[L ≤ α1L1 + α2L2] = 1

.

Let p = 1− p1 − p2 + q. Then, on the one hand, the VaR constraint becomes

P[L ≤ p1α1L1 + p2α2L2 − τWB

]= 1− p1 − p2 + q ,

as there is no event occurring with probability strictly lower than 1 − p1 − p2 + q.

On the other hand, we must consider P[0 ≤ L < α1L1] = 1− p1 − p2 + q. Because

every cumulative distribution function is strictly increasing, the condition

p1α1L1 + p2α2L2 − τWB < α1L1

must be satisfied. With L1 ≡ ` · L ≡ ` · (D + WB), we obtain the result shown in

Table 3.6.


Now let p = 1− p2: There are two ways to express this probability in terms of loan

redemptions, namely by P[L ≤ α1L1] and by P[L < α2L2], respectively. Hence,

there are two possibilities to express the constraint the bank faces, either as a weak

inequality,

p1α1L1 + p2α2L2 − τWB ≤ α1L1 ,

or as a strict inequality,

p1α1L1 + p2α2L2 − τWB < α2L2 ,

where both are shown in Table 3.6, again in terms of the total loan volume L.

Note that for all p < 1− p2, the constraint derived for p = 1− p1− p2 + q is binding

as otherwise the level of confidence p would drop to 1− p2 or to values over 1− p2.

The arguments from above hold analogously for α1L1 = α2L2 and α1L1 > α2L2,

respectively. Irrespective of the relative magnitude of both loan redemptions to each

other, we can derive the constraint for p = 1− q, resulting in

p1α1L1 + p2α2L2 − τWB < α1L1 + α2L2 ,

which is trivially fulfilled if L1, L2,WB ≥ 0 holds with at least one value being

strictly positive (otherwise equality holds).


Proof. By assumption, regulation is binding and implies Case 1 with `V = α2

α1+α2.

Thus, [v(`V , p) + ε(`V , p)

]·[Du(`V , RD; 1) +WB

] != τ ·WB

must be fulfilled. As the household’s initial wealth is not assumed to be binding

either, we can consider the unconstrained deposit-supply function. By (3.3.46), we

obtain v(`V , p) = (p1 + p2 − q) · α2

α1+α2and ε(`V , p) = ε. This equation yields two

solutions for RD whereas one solution always exceeds the other. Thus, the optimal

interest rate for the bank is unique, recall Result 5. The optimal interest rate is

equal to

RVD(1) =

q[ε+(p1+p2−1)

α1α2α1+α2

]+q(p1+p2−2q)

[2τ−ε−(p1+p2−1)

α1α2α1+α2

]α1α2α1+α2

γWB −√QV1 (`V ,p)

2q(1−q)[τ−ε−(p1+p2−1)

α1α2α1+α2

]γWB


where

QV1 (`V , p) = q2

[ε+ (p1 + p2 − 1) α1α2

α1+α2

]2

+

+ 2q[ε+ (p1 + p2 − 1) α1α2

α1+α2

]·

·q(p1+p2−2q)

[ε+

(p1+p2−1)α1α2α1+α2

]α1α2α1+α2

−2[(1−q)Rf−

(p1+p2−2q)α1α2α1+α2

][τ−ε− (p1+p2−1)α1α2

α1+α2

]γWB

+(p1 + p2 − 2q)

(p1+p2−2q)q[ε+

(p1+p2−1)α1α2α1+α2

]2−4(1−p1−p2+q)τ

[τ−ε− (p1+p2−1)α1α2

α1+α2

] α21α

22γ

2W 2B

(α1+α2)2

The regulatory constraint results in Formula (3.3.52) due to `V = α2

α1+α2. As DV

goes to zero for WB going to zero and as `V is independent of WB, the deposit

interest rate must go to RD( α2

α1+α2; 1). This limit

limWB↓0

RVD(1)

can be derived by l’Hopital’s rule. Taking once the derivative of the numerator and

the denominator with respect to WB and setting then WB = 0 yields

limWB↓0

RVD(1) =

1

qRf −

p1 + p2 − 2q

q

α1α2

α1 + α2

(3.2.11)∈ (

Rf − 1

q,1

qRf ) , i .e.

limWB↓0

RVD(1)

(3.2.55)= RD(

α2

α1 + α2

; 1) .

The signs for the sensitivities of RVD w/r/t WB, γ, Rf and τ can be obtained by

total differentiation of DV . Consider γ first.

dDV

dγ︸︷︷︸=0

≡ ∂Ds(`V , RVD; 1)

∂γ︸︷︷︸<0

+∂Ds(`V , RV

D; 1)

∂`· d`

V

dγ︸︷︷︸=0

+∂Ds(`V , RV

D; 1)

∂RD︸︷︷︸>0 by Res. 5

·dRVD

dγ,

resulting indRV

D

dγ> 0 .

Analogously, the sign w/r/t Rf obtains. As RVD is a function of γ ·WB,

dRVD

dWB

> 0

holds as well. Concerning τ ,

dDV

dτ︸︷︷︸>0

≡ ∂Ds(`V , RVD; 1)

∂τ︸︷︷︸=0

+∂Ds(`V , RV

D; 1)

∂`· d`

V

dτ︸︷︷︸=0

+∂Ds(`V , RV

D; 1)

∂RD︸︷︷︸>0 by Res. 5

·dRVD

dτ,


holds, so thatdRVDdτ

> 0 must hold.


Proof. Since `V = 12

is assumed to be optimal, the conditions on Case 1, C1, are

not violated and furthermore it is assumed that a potential maximizer to Problem

(3.3.44) does not lie at the border ∂C1. Moreover, the household is assumed not

to be constrained by its initial wealth WH and that p = 1 − 2p + q is the level

of confidence to be maintained, altogether implying that the bank’s Maximization

Problem (3.3.44) can be simplified to

max`, RD, j

E[WB(`, RD; j)

]

s.t. [v(`, p) + ε(`, p)] · [Du(`, RD; j) +WB] ≤ τ ·WB

where the constraint function, v(`, p) + ε(`, p), is fixed to v(12, 1− 2p+ q) + ε(1

2, 1−

2p + q) ≡ (p− 12)α + ε as the potential maximizer `V = 1

2lies at the kink of v(`, ·)

in `.

Then a maximum (12, RV

D, 1) satisfies the Kuhn-Tucker conditions and there is a

multiplier λ ≥ 0 such that 4

∂E[WB(`V ,RVD,1)]

∂RD= λ ·

[v(1

2, p) + ε(1

2, p)]· ∂Du(`V ,RVD,1)

∂RD

0 = λ ·[v(1

2, p) + ε(1

2, p)]·[Du(`V , RV

D, 1) +WB

]− τ ·WB

holds. Because regulation is assumed to be binding, we obtain λV > 0 implying

DV =[2(τ − ε)− (2p− 1)α]WB

(2p− 1)α + 2ε

and the identity Du(`V , RVD, 1) ≡ DV leads to

RVD =

q·[(2p−1)α+2ε]·[1−(p−q)αγWB ] + 4q(p−q)αγ·τWB −√q·QV (1−2p+q)

2q(1−q)[2(τ−ε)−(2p−1)α]γWB

4As ` = 12 maximizes the bank’s objective if both firms are equal when there is no regulation,

too, it does not change results irrespective whether we take the derivative of the bank’s objectivefunction with respect to ` or not given that the regulatory constraint is fixed to v( 1

2 , p) + ε( 12 , p).



QV (1− 2p+ q) = q (2p− 1)α + 2ε+ (p− q)α [2(2τ − ε)− (2p− 1)α] γWB2

− 4(1− q)γWB [2(τ − ε)− (2p− 1)α] ·· [(2p− 1)α + 2ε] [Rf − (p− q)α] + (p− q)(1− 2p+ 2q)α2γτWB .

The partial derivative of RVD with respect to WB can be represented as

∂RVD∂WB

= [(2p−1)α+2ε]√q·QV (1−2p+q)

·

·q·[(2p−1)α+2ε]+2q(p−q)α·ε+4(τ−ε)[(p−q)α−(1−q)Rf ]+(2p−1)[2(1−q)Rf−(2−q)(p−q)α]α·γWB

2(1−q)[2(τ−ε)−(2p−1)α]γW 2B

−√q·QV (1−2p+q)

2(1−q)[2(τ−ε)−(2p−1)α]γW 2B

which is strictly positive if and only if

q · [(2p− 1)α + 2ε] +

+ 2q(p− q)α · ε+ 4(τ − ε) [(p− q)α− (1− q)Rf ] +

+(2p− 1) [2(1− q)Rf − (2− q)(p− q)α]α γWB

2>

>√q ·QV (1− 2p+ q)

2

holds. Expanding both sides of the inequality and then subtracting the term of the

right-hand side from the term of the left hand side leads to

−4(1− q) [2(τ − ε)− (2p− 1)α]︸︷︷︸>0

γ2W 2B ·

·

− [2(τ − ε)− (2p− 1)α] ·

(p− q)α(α−Rf ) + [(1− q)Rf − (p− q)α]Rf︸︷︷︸

>0

︸︷︷︸<0

+ (p− q)α · −(2p− 1)(1− p+ q)α2 + 2qRfε

+ [(2− 2p+ q)(τ − ε) + q(2p− 1)Rf − qε]α> 0

As the factor in the first line is negative, the second, embraced factor must be

negative, too. As there are negative as well as positive expressions contained in the

latter, the following condition on τ ,

τ ≥[(2p− 1)α + 2ε]

[(p− q)2α2 − (p− q)(2− q)αRf + (1− q)R2

f

]

(p− q)(2p− q)α2 − 4(p− q)αRf + 2(1− q)R2f


makes the second factor negative, too, implying

∂RVD

∂WB

> 0 .


Appendix B

Derivations to Chapter 5

B.1 Bernoulli Mixture Model

In this section, we present a fairly general formulation of a Bernoulli mixture model.1

A Bernoulli mixture model is a model of Bernoulli-distributed random variables by

which various patterns and degrees of dependence among the Bernoulli random

variables can be accounted for. This is done by assuming the success probabilities

to be random themselves. Finally, we will derive the first and all second moments for

this model under the assumption that the success probabilities p(j)i follow (5.2.2). For

convenience, we dispense with the cases j = 1 and j = n and treat the probabilities

p(j)i as if they were equally given across all firms j within a given sector i. This is

further justified by the fact that we aim at providing the moments of the limiting

distribution of aggregate loan redemption in the next section, Section B.2.

The (cumulative) distribution functions of the probabilities p(j)i are denoted by

Fi(p(j)i ) with support on [0, 1], or a subarea thereof.

Realizations of the random variable p(j)i are denoted by p

(j)i . Given realizations p

(j)i ,

the Bernoulli random variables X(j)i are conditionally independent.

First, let us calculate the probability of a given outcome X(j)i j=1,...,n = x(j)

i ,x

(j)i ∈ 0, 1, within a given sector i. This probability is given by (cf. Joe, 2001,

pp. 211, 219)

P[X

(j)i = x

(j)i , j = 1, . . . , n

]=

∫

pi∈[0,1]n

n∏

j=1

(p(j)i )x

(j)i · (1− p(j)

i )1−x(j)i dF

(1,...,n)i (pi),

(B.1.1)

1Expositions on general properties of (Bernoulli) exchangeable mixture models can be found inJoe (2001, pp. 211, 219-220) and in Frey/McNeil (2001, pp. 7-8).

297

298 APPENDIX B. DERIVATIONS TO CHAPTER 5

where pi = (p(j)i )j=1,...,n and where F

(1,...,n)i (pi) denotes the joint distribution function

of pi = (p(j)i )j=1,...,n. The same formula applies to any subset of kn random variables

considered where 1 ≤ kn < n.

Let us extend the model to calculate the probability of the joint outcome

(X(j)1 , X

(j)2 )j=1,...,n = (x(j)

1 , x(j)2 )j=1,...,n of both sectors. Let F 1,...,n

1,2 (p1,p2)

be the common distribution function of the bivariate tuple (p1, p2) of success

probability vectors. Then the probability of a given drawing (X(j)1 , X

(j)2 )j=1,...,n =

(x(j)1 , x

(j)2 )j=1,...,n is computed according to

P[X

(j)i = x

(j)i , j = 1, . . . , n, i = 1, 2

]=

=

∫

(p1,p2)∈[0,1]2n

2∏

i=1

(n∏

j=1

(p(j)i )x

(j)i · (1− p(j)

i )1−x(j)i

)dF 1,...,n

1,2 (p1,p2) (B.1.2)

because of the conditional independence between all the random variables

considered. Likewise, the probabilities of kn tuples, 1 ≤ kn < n, can be calculated.

According to (B.1.1), the expected value of the random variable p(j)i is equivalent to

the success probability of the Bernoulli-distributed return X(j)i ,

P[X

(j)i = 1

]=

∫

p(j)i

p(j)i dFi(p

(j)i ) = E[p

(j)i ] = p

(j)i . (B.1.3)

Then the expected value of X(j)i is given by

E[X(j)i ] = E[p

(j)i · 1 + (1− p(j)

i ) · 0] = p(j)i (B.1.4)

because of (B.1.3). Likewise, we obtain for the variance

V[X(j)i ] = E

[(X

(j)i

)2]−(p

(j)i

)2

= E[p

(j)i · 12

]−(p

(j)i

)2

= p(j)i · (1− p(j)

i ) .

(B.1.5)

As all Bernoulli random variables considered either take zero or one as values, all

non-central moments can be presented by the respective joint success probabilities

B.1. BERNOULLI MIXTURE MODEL 299

given by (B.1.1) and (B.1.2):2

E[X(j)i ] = P[X

(j)i = 1] ,

E[X(j)i · X(j+1)

i ] = P[X(j)i = X

(j+1)i = 1] ,

...

E[X(1)i · X(2)

i · . . . · X(n)i ] = P[X

(j)i = 1, j = 1, . . . , n] ,

E[X(j)1 · X(j)

2 ] = P[X(j)1 = X

(j)2 = 1] ,

E[X(j)1 · X(j+1)

2 ] = P[X(j)1 = X

(j+1)2 = 1] ,

...

(B.1.6)

Thus, covariance terms are given by

Cov(X(j)i , X

(k)h ) = P[X

(j)i = X

(k)h = 1] − p

(j)i · p(k)

h , i, h = 1, 2, j = 1, . . . , n .

(B.1.7)

Consider the factor model for the success probabilities according to (5.2.2),

p(j)i =

1− bi · ξ(1)i − ci · ξ(2)

i − di · ξ(1)3−i , j = 1

1− ai · ξ(j−1)i − bi · ξ(j)

i − ci · ξ(j+1)i − di · ξ(j)

3−i , j = 2, . . . , n− 1

1− ai · ξ(n−1)i − bi · ξ(n)

i − di · ξ(n)3−i , j = n

for i = 1, 2 where ai, bi, ci, di are fixed, strictly positive numbers and where ξ(j)i are

mutually independent, uniformly distributed random variables with ξ(j)i ∼ U(0, 2t),

t > 0, for all i = 1, 2, j = 1, . . . , n. Furthermore, Condition (5.2.3) be fulfilled.

The univariate density function of each p(j)i , j = 1, . . . , n− 1, can be traced back to

the mutually independent risk factors ξ(j)i involved and, thus, is given by

fi(p(j)i ) = f ∗4(ξ

(j−1)i , ξ

(j)i , ξ

(j+1)i , ξ

(j)3−i) =

1

(2t)4if 0 ≤ ξ

(j−1)i , ξ

(j)i , ξ

(j+1)i , ξ

(j)3−i ≤ 2t

0 else,

(B.1.8)

for 2 ≤ j ≤ n − 1. Dependent on the lag between the Bernoulli random variables,

the univariate density functions f j;j+1i (pji , p

j+1i ), f j;j+2

i (pji , pj+2i ), and the bivariate

density functions f j;j+11,2 (p

(j)1 , p

(j−1)2 ), f j;j+1

1,2 (p(j)1 , p

(j)2 ), and f j;j+1

1,2 (p(j)1 , p

(j+1)2 ) are of

interest, too. They can be traced back to the associated factor loadings ξ(j)i and ξ

(k)h

analogous to fi(p(j)i ).

2All these 2n−1 joint moments alternatively characterize the multivariate Bernoulli distribution,cf. Joe (2001, p. 211), Maydeu-Olivares/Joe (2005, p. 1010).


Following (B.1.3) and (B.1.4) we obtain for the expectations:

E[X(j)i ] = P[X

(j)i = 1] = E[p

(j)i ] = 1− (ai + bi + ci + di) · t , 2 ≤ j ≤ n− 1 ,

and in the same manner V(X(j)1 ) via (B.1.5).

By (B.1.1) and (B.1.6), the second joint, non-central moment E[X(j)i X

(j+1)i ] is given

by

E[X(j)i X

(j+1)i ] =

=

∫

p(j)i ,p

(j+1)i

p(j)i · p(j+1)

i · fi(p(j)i , p

(j+1)i ) d(p

(j)i , p

(j+1)i )

=

∫

[0,2t]6

p(j)i · p(j+1)

i · f ∗6(ξ(j−1)i , ξ

(j)i , ξ

(j+1)i , ξ

(j+2)i , ξ

(j)3−i, ξ

(j+1)3−i ) d(ξ

(j−1)i , . . . , ξ

(j+1)3−i )

= p2i +

1

3· (ai + ci) · bi · t2 , 2 ≤ j ≤ n− 1 , (B.1.9)

resulting by (B.1.7) in

Cov(X(j)i , X

(j±1)i ) =

1

3· (ai + ci) · bi · t2 , 2 ≤ j ≤ n− 1 . (B.1.10)

By this 2-dependence structure, any second mixed moment of X(j)i and X

(j+m)i with

lag |m| ≥ 3 becomes

E(X(j)i X

(j+m)i ) = E(X

(j)i ) · E(X

(j+m)i ) , 1 ≤ j +m ≤ n , (B.1.11)

within each sector, and any second mixed moment of X(j)i and X

(j+m)3−i with lags of

|m| ≥ 2

E(X(j)i X

(j+m)i ) = E(X

(j)i ) · E(X

(j+m)3−i ) , 1 ≤ j +m ≤ n , (B.1.12)

across sectors. Thus, the other intra- and inter-sectoral covariance terms can be

determined. The corresponding correlations are given by (5.2.6) to (5.2.10).

B.2. MOMENTS OF AGGREGATE LOAN REPAYMENTS 301

B.2 Moments of Aggregate Loan Repayments

Let us determine the expectations, variances and covariances of the sector-wise

aggregated loan portfolios

L1 =n∑

j=1

X(j)1 · `1 and L2 =

n∑

j=1

X(j)2 · `2 ,

where we treat all Bernoulli random variables, X(j)i , as homogeneous within each

sector i. That is, we neglect the asymmetries arising at the borders of the sequence

X(1)i , X

(2)i , . . . , X

(n−1)i , X

(n)i . Note that we are interested in the statistical moments

of large portfolios, that is with many borrowers, and in the statistical moments of

the associated limiting distribution such that this approach may be justified.

The expectations are thus simply given by

E

[n∑

j=1

X(j)i · Li

]= n · piαi ·

Lin

= piαi · Li .

The intra-sectoral variance becomes

V( n∑

j=1

X(j)i · Li

)=

n∑

j=1

V(X(j)i · Li) +

n∑

j=1

Cov(X

(j)i · Li,

n∑

k=1,k 6=j

X(k)i · Li

)

=n∑

j=1

V(X(j)i ) · L2

i + Cov(X(1)i , X

(n)i ) · L2

i +n∑

j=2

Cov(X(j)i , X

(j−1)i ) · L2

i

+n−1∑

j=1

Cov(X(j)i , X

(j+1)i ) · L2

i + Cov(X(n)i , X

(1)i ) · L2

i

+ Cov(X(1)i , X

(n−1)i ) + Cov(X

(2)i , X

(n)i ) +

n∑

j=3

Cov(X(j)i , X

(j−1)i )

+n−2∑

j=1

Cov(X(j)i , X

(j+2)i ) · L2

i + Cov(X(n−1)i , X

(1)i ) · L2

i + Cov(X(n)i , X

(2)i ) · L2

i

= n · V(X(1)i ) · L2

i + 2 · n · Cov(X(1)i , X

(2)i ) · L2

i + 2 · n · Cov(X(1)i , X

(3)i ) · L2

i

as all covariance terms with a distance of |m| ≥ 3 reduce to zero according to

(B.1.11). Hence, the variance of each aggregate loan repayment Li is finally given


by

V(Li) =1

n·[V(X

(1)i ) + 2 · Cov(X

(1)i , X

(2)i ) + 2 · Cov(X

(1)i , X

(3)i )]· L2

i . (B.2.1)

Due to Property (B.1.12), the covariance between the sums∑n

j=1 X(j)1 · `1 and∑n

j=1 X(j)2 · `2 is given by

Cov( n∑

j=1

X(j)1 · `1,

n∑

j=1

X(j)2 · `2

)=

n∑

j=1

Cov(X

(j)1 · `1,

n∑

k=1

X(k)2 · `2

)

=

[Cov(X

(1)1 , X

(n)2 ) +

n∑

j=2

Cov(X(j)1 , X

(j−1)2 ) +

n∑

j=1

Cov(X(j)1 , X

(j)2 )

]· `1 · `2

+n−1∑

j=1

Cov(X(j)1 , X

(j+1)2 ) · `1 · `2 + Cov(X

(n)1 , X

(1)2 ) · `1 · `2

=[Cov(X

(2)1 , X

(1)2 ) + Cov(X

(1)1 , X

(1)2 ) + Cov(X

(1)1 , X

(2)2 )]· n · `1 · `2 ,

hence

Cov(L1, L2) =1

n·[Cov(X

(2)1 , X

(1)2 ) + Cov(X

(1)1 , X

(1)2 ) + Cov(X

(1)1 , X

(2)2 )]·L1 ·L2 .

(B.2.2)

In light of the Bernoulli mixture model from Sections B.1 and 5.2, the moments

of the aggregate loan repayments are given in terms of the factor loadings and the

individual expected success probability pi as follows:

E(Li) = piαi · Li , (B.2.3)

V(Li) =1

n·[pi(1− pi) +

2

3(aibi + aici + bici)t

2

]· α2

i · L2i . (B.2.4)

The covariance becomes

Cov(L1, L2) =1

3 · n · [(a1 + b1 + c1) · d2 + (a2 + b2 + c2) · d1] · t2 · α1 · α2 · L1 · L2 ,

(B.2.5)

resulting in

Corr(L1, L2) =13· [(a1 + b1 + c1) · d2 + (a2 + b2 + c2) · d1] · t2

2∏i=1

√pi(1− pi) + 2

3(aibi + aici + bici)t2

(B.2.6)

as inter-sectoral correlation.

Appendix C

Proofs to Chapter 6

C.1 Proof of Result 30

Proof. To prove the existence of a solution to the household’s unconstrained

maximization problem it is crucial to show that

limD→±∞

−e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2] ·[Φ(µ−γσ

2(D+WB)σ

)− Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

]= −∞

holds. The first term, −e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2], diverges for D −→ ±∞to −∞ whereas the second term converges from above to zero. To determine the

limit we use de l’Hopital’s rule. Therefore we consider the limit of

limD→±∞

− Φ(µ−γσ2(D+WB)σ

)− Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

(e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2]

)−1 . (C.1.1)

The derivative of the second term (the numerator) is

−γσ · ϕ(µ−γσ2(D+WB)σ

) + (γσ + RDWB

σ(D+WB)2) · ϕ(

µ− DRDD+WB

−γσ2(D+WB)

σ) > 0 .

Taking the derivative of the denominator with respect to D and summarizing

exponential factors yields:

−(e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2]

)−1

·[−γ(µ−Rf )− 2γ2σ2(D +WB)

]> 0 .

303

304 APPENDIX C. PROOFS TO CHAPTER 6

Let us define

E1 = e

[− µ2

2σ2−γWHRf+γDRf+ 12γ2σ2(D+WB)2

]

,

and

E2 = e

[−γDRD+ 1

σ2 ·DRDD+WB

·(µ− DRD2(D+WB)

)].

Their limiting behavior is given by

limD→±∞

E1 = +∞ , limD→−∞

E2 = +∞ , and limD→+∞

E2 = 0 . (C.1.2)

Simplifying the ratio of these two derivatives by using the just defined terms E1 and

E2 results in:

γσ · E1 ·[

1 + RDγσ2(D+WB)2

]· E2 − 1

√2π · [γ(µ−Rf )− 2γ2σ2(D +WB)]

.

Both the numerator and the denominator diverge for D −→ ±∞, but each with

opposite sign. More precisely, we obtain for the numerator:

limD→+∞

γσ · E1 ·[

1 +RD

γσ2(D +WB)2

]· E2 − 1

= −∞

limD→−∞

γσ · E1 ·[

1 +RD

γσ2(D +WB)2

]· E2 − 1

= +∞

In either case, the rule of de l’Hopital can potentially be applied once more. The

derivative of the denominator is simply −2√

2π · γ2σ2 such that the behavior of the

numerator remains crucial.

The derivative with respect to D of the numerator reads:

γσ · E1 ·[

1 + RDγσ2(D+WB)2

]· E2 − 1

· [γRf + γ2σ2 · (D +WB)]

+ γσ · E1 ·[

1− 2RDγσ2(D+WB)3

]· E2 + . . .

. . .+[1 + RD

γσ2(D+WB)2

]·[−γRD + 1

σ2 · RDWB

(D+WB)2·(µ− DRD

D+WB

)]· E2 − 1

Recall (C.1.2). Thus, for D −→ +∞, the numerator becomes

+ ∞ · [1 + 0] · 0− 1 · [+∞]

+ ∞ · [1− 0] · 0 + [1 + 0] · [−γRD + 0 · (µ−RD)] · 0− 1= ∞ · 0− 1 · ∞+∞ · 0 + 0− 1 = −∞ .

Since the derivative of the denominator is equal to −2√

2π · γ2σ2, the whole ratio

C.1. PROOF OF RESULT 30 305

diverges for D −→ +∞ to +∞. By de l’Hopital’s rule this holds true for both

preceding ratios and especially we obtain

limD→+∞

−e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2]·[Φ(µ−γσ

2(D+WB)σ

)− Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

]= −∞

because we have omitted the sign of Ratio (C.1.1).

Considering D −→ −∞ yields:

+ ∞ · [1 + 0] · ∞ · [−∞]

+ ∞ ·

[1 + 0] · ∞ + [1 + 0] ·[−γRD + 1

σ2 · 0 · (µ−RD)]· ∞ − 1

= ∞ · ∞ · [−∞] + ∞ · [1− γRD] · ∞ = −∞ ,

as the first term, that is negative, outweighs the second term, that is positive, by

an order of magnitude of D (the first term grows at E1 · E2 ·D whereas the second

term only by an order of magnitude of E1 · E2). Thus we have,

limD→−∞

γσ · E1 ·[

1 +RD

γσ2(D +WB)2

]· E2 − 1

[γRf + γ2σ2(D +WB)] = −∞ ,

which in fact is irrespective of the sign of the second term. To summarize, we obtain

the same result as for D −→ +∞ and hence

limD→±∞

−e−γ[µ(D+WB)+(WH−D)Rf−γσ2(D+WB)2]·[Φ(µ−γσ

2(D+WB)σ

)− Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

]= −∞ .

(C.1.3)

The remainder of the expected utility function (6.3.1) becomes for D −→ −∞

limD→−∞

−e−γ(WH−D)Rf · Φ(−µσ

) = 0 ,

limD→−∞


D+WB

σ) = −∞ ,


and for D −→ +∞

limD→+∞

−e−γ(WH−D)Rf · Φ(−µσ

) = −∞ ,

limD→+∞


D+WB

σ) = 0 ,

where RD > Rf has been assumed to derive the limits of that part of expected

utility that appreciates the state of full deposit redemption. Any other other relation

between RD and Rf does not change the overall result, however.

In the end, we have shown that

limD→±∞

E[uH(WH)] = −∞ (C.1.4)

holds. Since E[uH(WH)] is continuous (and differentiable) for all real D, there is

always at least one real number Du maximizing E[uH(WH)]. By strict concavity of

E[uH(WH)] with respect to D, Du is unique.


Proof.

The Lower Bound to the Unconstrained Deposit Supply Function

Consider the partial derivative of expected utility with respect to D which is given


by:

∂E[uH(WH)]∂D

= − e−γ(WH−D)Rf · Φ(−µσ) · γRf

− e−γ[µ(D+WB)+(WH−D)Rf− 12γσ2(D+WB)2] ·

·[Φ(µ−γσ

2(D+WB)σ

) − Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

]·

· (−γ) · [µ−Rf − γσ2(D +WB)]


·[ϕ(µ−γσ

2(D+WB)σ

) · (−γσ) −

− ϕ(µ− DRD

D+WB−γσ2(D+WB)

σ) · 1

σ·(− WBRD

(D+WB)2− γσ2

)]

− e−γ[DRD+(WH−D)Rf ] · Φ(µ− DRD

D+WB

σ) · (−γ) · (RD −Rf )

− e−γ[DRD+(WH−D)Rf ] · ϕ(µ− DRD

D+WB

σ) · (− 1

σ) · WBRD

(D+WB)2

Gathering all terms involving exponentials, in particular including those with the

density function of the normal distribution ϕ(·), yields

∂E[uH(WH)]∂D

= − e−γ(WH−D)Rf · Φ(−µσ) · γRf


·[Φ(µ−γσ

2(D+WB)σ

) − Φ(µ− DRD

D+WB−γσ2(D+WB)

σ)

]·

· (−γ) · [µ−Rf − γσ2(D +WB)]

− e−γ[DRD+(WH−D)Rf ] · Φ(µ− DRD

D+WB

σ) · (−γ) · (RD −Rf )

+ e−γ(WH−D)Rf · ϕ(µσ) · γσ

− e−γ[DRD+(WH−D)Rf ] · ϕ(µ− DRD

D+WB

σ) · γσ ,

Dividing by γe−γ(WH−D)Rf , the first-order condition becomes

− e−γDRD · σ · ϕ(µ− DRD

D+WB

σ) + σ · ϕ(µ

σ)

+ e−γ·[µ(D+WB)− 12γσ2(D+WB)2] · [Rf − µ+ γσ2(D +WB)] ·∆Φ

− Rf · Φ(−µσ) + e−γDRD · (RD −Rf ) · Φ(

µ− DRDD+WB

σ) = 0 ,

(C.2.1)


where

∆Φ =

[Φ(µ− DRD

D+WB− γσ2(D +WB)

σ)− Φ(

µ− γσ2(D +WB)

σ)

], (C.2.2)

hence, −1 < ∆Φ < 0 holds. By (5.4.14), the first-order condition may be

approximated by


D+WB

σ)

+ e−γ·[µ(D+WB)− 12γσ2(D+WB)2] ·

[Rf − µ+ γσ2(D +WB)

]·∆Φ

+ e−γDRD · (RD −Rf ) · Φ(µ− DRD

D+WB

σ) = 0 .

Let

D =µ−Rf

γσ2− WB

be a solution to the household’s problem. Hence, Rf −µ+ γσ2(D+WB) ≡ 0 holds.

The approximated first-order condition becomes at D = D


D+WB

σ)

e−γDRD · (RD −Rf ) · Φ(µ− DRD

D+WB

σ) > 0 ,

whereas the positive sign is due to Assumption (6.3.4). Thus, in conjunction with

strict concavity and hence uniqueness, as established by Result 30, the household

can improve unambiguously by increasing D. Hence, the optimal unconstrained

deposit supply satisfies

Du >µ−Rf

γσ2− WB .

Behavior of the Deposit Supply Function in Rf

Consider the household’s first-order condition


D+WB

σ) + σ · ϕ(µ

σ)

+ e−γ·[µ(D+WB)− 12γσ2(D+WB)2] · [Rf − µ+ γσ2(D +WB)] ·∆Φ

− Rf · Φ(−µσ) + e−γDRD · (RD −Rf ) · Φ(

µ− DRDD+WB

σ)

!= 0 .

(C.2.3)

Let LHSFOC be the left-hand side of this equation and by definition LHSFOC =


∂E[WH(`,RD)]∂D

· 1γ· eγ(WH−D)Rf holds.

Since we have ∂2E[uH(WH)]∂D2 < 0 by strict concavity, the signs of the partial derivatives

∂D∂θ

are exclusively determined by the signs of ∂2E[uH(WH)]∂θ∂D

where θ is the respective

parameter considered.

Let us define

A =

[ϕ(µ

σ)− e−γDRDϕ(

µ− DRDD+WB

σ)

]< 0 ,

B = e−γ·[µ(D+WB)−γσ2(D+WB)2] ·∆Φ < 0 ,

where the sign of expression A is due to Approximation (5.4.14), i.e. ϕ(µσ) ≈ 0.

Consider now the second mixed partial derivative of LHSFOC with respect to Rf ,

which is simply given by

∂LHSFOC

∂Rf

= B − Φ(−µσ

) − e−γDRD · Φ(µ− DRD

D+WB

σ) < 0 .

As

∂2E[uH(WH)]

∂Rf∂D=

∂LHSFOC

∂Rf︸︷︷︸<0

·1γ· eγ(WH−D)Rf + LHSFOC︸︷︷︸

≡0

·1γ· eγ(WH−D)Rf · (−γ ·D)

holds,∂2E[uH(WH)]

∂Rf∂D< 0

obtains. Thus, the unconstrained deposit supply strictly decreases in Rf .

Behavior of the Deposit Supply Function in µ and σ

To determine the sign of ∂D∂µ

we must analyze ∂2E[uH(WH)]∂µ∂D

which is equal to ∂LHSFOC

∂µ

except for constants. Thus this second mixed derivative is proportional to:

∂2E[uH(WH)]∂µ∂D

∼ −γσ · (D +WB) · A

+ WB ·RDσ(D+WB)

· e−γDRD · ϕ(µ− DRD

D+WB

σ)

− 1 + [Rf − µ+ γσ2 · (D +WB)] · γ(D +WB) ·B(C.2.4)

We know from above A < 0 and B < 0. The expression [Rf − µ+ γσ2(D +WB)] is


positive because of (6.3.5). So we obtain

∂2E[uH(WH)]

∂µ∂D> 0 , and thus

∂Du(`, RD)

∂µ> 0 . (C.2.5)

Next we consider the partial derivative with respect to σ in an analogous way:

∂2E[uH(WH)]∂σ∂D

∼ 1 + [Rf + γσ2(D +WB)] γ(D +WB) · A+ γσ2DRD(D+WB)2−RDWB [µ(D+WB)−DRD]

σ2(D+WB)2· e−γDRD · ϕ(

µ− DRDD+WB

σ)

+ 2 + [Rf − µ+ γσ2(D +WB)] · γ(D +WB) · γσ(D +WB) ·B(C.2.6)

Approximating ∂2E[uH(WH)]∂σ∂D

by ϕ(µσ) ≈ 0 yields

∂2E[uH(WH)]∂σ∂D

≈

−1− [Rf + γσ2(D +WB)] γ(D +WB) + γDRD − [µ(D+WB)−DRD]·WB ·RD

σ2(D+WB)2

· . . .

. . . · e−γDRD · ϕ(µ− DRD

D+WB

σ)

+ 2 + [Rf − µ+ γσ2(D +WB)] · γ · (D +WB) · γσ · (D +WB) ·B(C.2.7)

The first term in curly braces is negative because

− [Rf + γσ2(D +WB)] · γ · (D +WB) + γDRD <

< − [Rf + µ−Rf ] · γ · (D +WB) + γDRD =

= −γ · [µ(D +WB)−DRD] < 0

holds by (6.3.5). The second term is negative for the same reasons as it is the case

in the last line of Derivative (C.2.4).


Proof. Here, the expression deposit supply always refers to the unconstrained

deposit supply on the domain [RD(`),∞).

Since we have ∂2E[uH(WH)]∂D2 < 0 by strict concavity, the sign of ∂D

∂RDis determined by


the sign of ∂2E[uH(WH)]∂RD∂D

. The latter reads:

∂2E[uH(WH)]∂RD∂D

= − D·WB ·RDσ(D+WB)2

· e−γDRD · ϕ(µ− DRD

D+WB

σ)

+ e−γDRD · Φ(µ− DRD

D+WB

σ)

− γD · (RD −Rf ) · e−γDRD · Φ(µ− DRD

D+WB

σ) .

(C.3.1)

That is, ∂2E[uH(WH)]∂RD∂D

is positive if and only if

RD <(1 + γ ·DRf ) · Φ(

µ− DRDD+WB

σ)

ϕ(µ− DRD

D+WB

σ) · D·WB

σ(D+WB)2+ γ ·D · Φ(

µ− DRDD+WB

σ)

(C.3.2)

holds. We define

f(RD) =(1 + γ ·DRf ) · Φ(

µ− DRDD+WB

σ)

ϕ(µ− DRD

D+WB

σ) · D·WB

σ(D+WB)2+ γ ·D · Φ(

µ− DRDD+WB

σ)

.

To draw conclusions about the existence and uniqueness of Ru

D(`), we will analyze

the behavior of f(RD) in RD.

We note that ∂D∂RD

exists on (RD(`),∞) by Result 31 and that

limRD↓RD(`)

f(RD) = ∞ (C.3.3)

holds.

The derivative of the numerator of f(RD) with respect to RD is,

∂ •∂RD

= γRf · ∂D∂RD· Φ(

µ− DRDD+WB

σ) + (1 + γDRf ) · ϕ(

µ− DRDD+WB

σ) · Ω (C.3.4)

where Ω is defined as

Ω ≡ − D

σ(D +WB)− RDWB

σ(D +WB)2· ∂D∂RD

. (C.3.5)

Ω < 0 holds for WB = 0 and Ω < 0 holds for WB > 0 if and only if

∂D

∂RD

> −D · (D +WB)

RD ·WB


RD = RDf (RD)RD = RDf (RD)

Figure C.1: Feasible graphs for the function f(RD)

This figure illustrates feasible graphs for the function f(RD) in relation to the identity RD ≡ RD.The left-hand chart summarizes the conclusions drawn by (C.3.2), (C.3.7), and (C.3.3). The right-hand chart shows generic graphs for f(RD) as implied by this proof.

is satisfied. The derivative of the denominator of f(RD) with respect to RD is:

∂ ∂RD

= − ϕ(µ− DRD

D+WB

σ) · D ·WB

σ(D +WB)2·µ− DRD

D+WB

σ· Ω

+ γ · ∂D∂RD

· Φ(µ− DRD

D+WB

σ) + γD · ϕ(

µ− DRDD+WB

σ) · Ω . (C.3.6)

After having expanded and simplified the numerator of f ′(RD) we obtain

ϕ(µ− DRD

D+WB

σ) · Φ(

µ− DRDD+WB

σ) · D ·WB

σ(D +WB)2· γ ·Rf ·

∂D

∂RD

+ ϕ(µ− DRD

D+WB

σ) · Φ(

µ− DRDD+WB

σ) · D ·WB

σ(D +WB)2·

(1 + γDRf ) ·(µ− DRD

D+WB

)

σ· Ω

− γ · Φ(µ− DRD

D+WB

σ)2 · ∂D

∂RD

+ ϕ(µ− DRD

D+WB

σ)2 · DWB

σ(D +WB)2· (1 + γDRf ) · Ω

Recall Condition (6.3.10), i.e.

ϕ(busz)

Φ(busz)· D

u ·WB ·Rf

σ(Du +WB)2< 1 ,

This condition allows the following two conclusions,

∂D

∂RD

> 0 ⇒ f ′(RD) < 0 ,

f ′(RD) > 0 ⇒ ∂D

∂RD

< 0 . (C.3.7)


For RD ≥ RD(`), ∂D∂RD

> 0 always holds, provided that RD is not “too big”, because

any agent, independent of his or her risk preferences, is ready to take some risk as

soon as the promised return RD exceeds the zero RD(`) which equals the expected

return on deposits for an ε→ 0 invested into deposits.

Thus, in the neighborhood of RD(`) with RD ≥ RD(`) , f(RD) strictly decreases

in RD and has a pole for RD approaching RD(`) from below. These two properties

base on (C.3.7) and (C.3.3), respectively. Furthermore, ∂D∂RD

> 0 implies via (C.3.2)

the relation RD < f(RD). The left-hand chart of Figure C.1 summarizes these

observations.

Assume now that f(RD) increases while remaining above the identity. Then deposit

supply decreases in RD, ∂D∂RD

< 0, according to (C.3.7). This, in turn, leads to

RD > f(RD) because of Relation (C.3.2), meaning that f(RD) is in fact located

below the identity: hence, a contradiction to our assumption. Consequently, f(RD)

decreases until it crosses the identity, implying by (C.3.2) that ∂D∂RD

< 0 holds

thereafter and thus Ru

D(`) exists.

RD = RDf (RD)

1.06

1.08

1.10

1.12

1.14

1.06 1.08 1.10 1.12 1.14

1.0010

1.0015

1.0020

1.0025

1.0030

1.0000 1.0005 1.0010 1.0015 1.0020 1.0025 1.0030

1.0005

RD = RDf (RD)

Figure C.2: The function f(RD) and its relation to the identity RD ≡ RD

This figure illustrates the course of the function f(RD). The left-hand chart shows the graphof f(RD) with the parameter values given in Table 6.1 and given ` = 0.597. The right-handchart shows the graph of f(RD) with the parameter values given in Table 6.2, Panel A and given` = 0.628. Hence, the chosen loan-allocation rates equal their respective laissez-faire equilibriumvalues. Cf. Figure A.1.

Let us establish uniqueness. Assume first that f(RD) crosses the identity a second

time from below. After this second intersection we have RD < f(RD) implying∂D∂RD

> 0 by (C.3.2). But we also have, at least in a neighborhood above that

intersection, f ′(RD) > 0 leading to a contradiction because of (C.3.7). Thus, after

this “intersection” we must obtain f ′(RD) < 0. That is, any potential second

intersection with the identity is in fact a tangency point. However, this tangency

point vanishes if the model is perturbed slightly. Thus, Ru

D(`) is generically unique.


The right-hand chart of Figure C.1 summarizes the idea of this proof.

Figure C.2 shows the relation between RD and f(RD) with the parameter values

given in Tables 6.1 and 6.2, Panel A. The loan-allocation rates ` equal their laissez-

faire equilibrium values, i.e. ` = 0.597 and ` = 0.628, respectively.

Appendix D

Proofs to Chapter 7

D.1 Existence of the Deposit Supply Function in

the First Period

Recall the household’s expected utility in t = 0 over final wealth. First of all,

∞∫

bdx,1

E1

[uH(WH,2) |WB,1

]· 1

σ1

· ϕ(z1) dx1 < 0

trivially holds for all D0 ∈ R where

z1 ≡x1 − µ1

σ1

(D.1.1)

is the centralized return on the whole loan portfolio after the first period and where

bdx,1 ≡D0RD,0

D0 +WB,0

(D.1.2)

is the bank’s solvency barrier after the first period in terms of the non-centralized

portfolio returns (cf. bdz ,t+1).

Second, the limits for the last term if D0 → ±∞ are given by

limD0→−∞

−eγD0Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

]= 0− ,

limD0→+∞

−eγD0Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

]= −∞ .

315

316 APPENDIX D. PROOFS TO CHAPTER 7

Let

∆Φ1 = Φ(µ2 − γσ2

2(D1 +WB,1)

σ2

)− Φ(µ2 − D1RD,1

D1+WB,1− γσ2

2(D1 +WB,1)

σ2

)

(D.1.3)

and analogously

∆Φ0 = Φ(µ1 − γσ2

1(D0 +WB,0)Rf,1

σ1

)− Φ(µ1 − bdx,1 − γσ2

1(D0 +WB,0)Rf,1

σ1

) .

(D.1.4)

Furthermore, for the last but second term one obtains

limD0→±∞

−e−γ[µ1(D0+WB,0)−D0Rf,0− 12γσ2

1(D0+WB,0)2Rf,1]Rf,1 ·∆Φ0 = −∞ ,

for the derivation of this limit, we refer to Appendix C.1.

Altogether, we obtain

limD0→±∞

E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]= − ∞ . (D.1.5)

This property results in conjunction with differentiability in D0 in at least one

extremal point. That is,

∂ E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]

∂D0

!= 0 (D.1.6)

has at least one solution on R in D0.

D.2 Proof of Result 36

If the second period’s choices, Ds1(·), R∗D,1(·), and `∗D,1(·), were constants, the

concavity in D0 of the household’s expected utility in t = 0 over final wealth WH,2

would follow immediately. However, as Ds1(·), R∗D,1(·) and `∗D,1(·) depend on D0 in

a non-trivial manner, the arguments presented so far, as done on p. 181 are not

enough.

In the next paragraph, we will characterize the second derivative of

E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]in each extremal point. By doing so, we obtain conditions

for the second derivative’s negative sign and hence uniqueness of D0 can be

D.2. PROOF OF RESULT 36 317

established. To keep notations under integrals short, WB,1 or realizations of it,

WB,1, will occasionally be skipped under the expectations operator and replaced by

a dot.

The first-order condition

We define

g(D0) := −∞∫

bdx,1

e−γ(WH,1−D1)Rf,1 ·

1− Φ(µ2

σ2

)

+ e−γ[µ2(D1+WB,1)− 12γσ2

2(D1+WB,1)2] ·∆Φ1

+ e−γD1RD,1 · Φ(bdz ,2)

· 1

σ1

· ϕ(z1) dx1

≡∞∫

bdx,1

E1

[uH(WH,2) |WB,1

]· 1

σ1

· ϕ(z1) dx1 (D.2.1)

and

h(D0) = −e−γ[µ1(D0+WB,0)+(WH,0−D0)Rf,0− 12γσ2

1(D0+WB,0)2Rf,1]Rf,1 ·∆Φ0

−e−γ(WH,0−D0)Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

](D.2.2)

such that

E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]≡ g(D0) + h(D0)

holds. Thus the household’s first-order condition in t = 0 can be represented by

g′(D0) + h′(D0) = 0 where

g′(D0) =

∞∫

bdx,1

(∂E1[uH(WH,2)|·]

∂WB,1

· (x1 −RD,0) +∂E1[uH(WH,2)|·]

∂WH,1

· (RD,0 −Rf,0)

+∂E1[uH(WH,2)|·]

∂RD,1

· ∂R∗D,1

∂D0

+∂E1[uH(WH,2)|·]

∂`1

· ∂`∗D,1

∂D0

)· 1

σ1

· ϕ(z1) dx1

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)


and

h′(D0) = γRf,1

[µ1 −Rf,0 − γσ2

1(D0 +WB,0)Rf,1

]·

· e−γ[µ1(D0+WB,0)+(WH,0−D0)Rf,0− 12γσ2

1(D0+WB,0)2Rf,1]Rf,1 ·∆Φ0

+ γσ1Rf,1e−γ(WH,0−D0)Rf,0 · ϕ(

µ1

σ1

)

−(γσ1Rf,1 +

RD,0WB,0

σ1(D0 +WB,0)2

)· e−γ[D0RD,0+(WH,0−D0)Rf,0]Rf,1 · ϕ(bdz ,1)

− γRf,0Rf,1 · eγ(WH,0−D0)Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

].

The derivative g′(D0) is obtained by the Leibniz rule and after some simplifying

algebraic manipulations. Note that

∂E1

[uH(WH,2) |WB,1

]

∂D1

≡ 0

holds given the equilibrium in t = 1. Consequently, the dependencies on D0 via

Ds1(·) vanish. The partial derivatives of expected utility E1[uH(WH,2)] have thus be

understood as the direct dependencies on the respective variables and parameters,

and never as dependencies via (equilibrium) deposit supply in t = 1.

Because of

∂E1[uH(WH,2) |WB,1 ]

∂WH,1

= − γRf,1 · E1

[uH(WH,2) |WB,1

]> 0 (D.2.3)

the derivative g′(D0) further simplifies to

g′(D0) =

∞∫

bdx,1

(∂E1[uH(WH,2)|·]

∂WB,1

· (x1 −RD,0) +∂E1[uH(WH,2)|·]

∂RD,1

· ∂R∗D,1

∂D0

+∂E1[uH(WH,2)|·]

∂`1

· ∂`∗D,1

∂D0

)· 1

σ1

· ϕ(z1) dx1

− γ ·Rf,1 · (RD,0 −Rf,0) · g(D0)

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) . (D.2.4)


We note that

limx1↓bdx,1

D1RD,1

D1 +WB,1

= 0 (D.2.5)

holds which can be illustrated by de l’ Hopital’s rule, yielding

limx1↓bdx,1

∂D1

∂x1·RD,1 + D1 · ∂RD,1∂x1

∂D1

∂x1+ (D0 +WB,0)

=∂D1

∂x1· 0 + 0 · ∂RD,1

∂x1

∂D1

∂x1+ (D0 +WB,0)

= 0

due to the Boundary Condition (7.1.3) and by additionally assuming that the

denominator will be unequal to zero generically. The partial derivatives do exist

from above.

Hence, we obtain

E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

= − e−γWH,1Rf,1 , (D.2.6)

as

limx1↓bdx,1

∆Φ1 = 0 (D.2.7)

because of (D.2.5). Thus, assuming RD,1 = 0 at x1 = bdx,1 instead of any other

value RD,1 ≤ RD,1 is not an arbitrary choice, but is sensible, as thus ∆Φ1 ≡ 0

obtains which is necessary to transform the expectations E1[uH(WH,2)]∣∣∣x1=bdx,1

into

a number independent of random drawings at the end of the second period. In

particular, final wealth is not only known in t = 1 to the household if the bank has

defaulted, but must also exactly equal the expression shown in (D.2.6).

At D0 = 0 we obtain

h′(0) = − RD,0

σ1WB,0

· e−γWH,0Rf,0Rf,1 · ϕ(µ1

σ1

)

− γRf,0Rf,1 · e−γWH,0Rf,0Rf,1 ·[1− Φ(

µ1

σ1

)

]< 0.

Because the function h(D0) is a linear combination of strictly concave functions in

D0, h(D0) is strictly concave in D0 which results in

h′(D0) < 0 ∀ D0 ≥ 0 . (D.2.8)

because of h′(0) < 0 by contradiction. Consequently,

g′(D0) > 0 (D.2.9)


holds in every extremal point.

The second-order condition

The second derivative of g(D0) is calculated by taking the following partial

derivatives with respect to D0:

g′′(D0) =d

dD0

∞∫

bdx,1

∂E1[uH(WH,2) |WB,1 ]

∂WB,1

· (x1 −RD,0) · 1

σ1

· ϕ(z1)dx1

+d

dD0

∞∫

bdx,1

∂E1[uH(WH,2) |WB,1 ]

∂RD,1

· ∂R∗D,1

∂D0

· 1

σ1

· ϕ(z1)dx1

+d

dD0

∞∫

bdx,1

∂E1[uH(WH,2) |WB,1 ]

∂`1

· ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

− γ ·Rf,1 · (RD,0 −Rf,0) · g′(D0)

+d

dD0

[− E1[uH(WH,2) |WB,1 ]

∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

].

Due to the first-order condition we know g′(D0) > 0 such that the last but one line

is negative provided RD,0 > Rf,0.


Taking the remaining partial derivatives and ordering them results in

− γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0)

+

∞∫

bdx,1

[∂2E1[uH(·)|·]

∂W 2B,1

· (x1 −RD,0)2 +∂2E1[uH(·)|·]

∂R2D,1

·(∂R∗D,1∂D0

)2

+

+∂2E1[uH(·)|·]

∂`21

·(∂`∗1∂D0

)2]· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

[∂2E1[uH(·)|·]∂WB,1∂WH,1

· (x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂WH,1

· ∂R∗D,1

∂D0

+

+∂2E1[uH(·)|·]∂`1∂WH,1

· ∂`∗1

∂D0

]· (RD,0 −Rf,0) · 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

[∂2E1[uH(·)|·]∂WB,1∂D1

(x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂D1

∂R∗D,1∂D0

+∂2E1[uH(·)|·]∂`1∂D1

∂`∗1∂D0

]·

·[∂Ds

1

∂D0

+∂Ds

1

∂RD,1

· ∂R∗D,1

∂D0

+∂Ds

1

∂`1

· ∂`∗1

∂D0

]· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 ·

(x1 −RD,0)

[∂2E1[uH(·)|·]∂WB,1∂`1

· ∂`∗1

∂D0

+∂2E1[uH(·)|·]∂WB,1∂RD,1

· ∂R∗D,1

∂D0

]

+∂2E1[uH(·)|·]∂RD,1∂`1

· ∂R∗D,1

∂D0

· ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

(x1 −RD,0)2

[∂E1[uH(·)|·]∂RD,1

∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

∂2`∗1∂W 2

B,1

]1

σ1

ϕ(z1)dx1

+d

dD0

[− E1[uH(·)|·]|x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

]

− RD,0WB,0

(D0 +WB,0)2

[∂E1[uH(·)|·]∂RD,1

∂R∗D,1∂D0

]∣∣∣∣x1=bdx,1

+

[∂E1[uH(·)|·]

∂`1

∂`∗1∂D0

]∣∣∣∣x1=bdx,1

− RD,0WB,0

D0 +WB,0

· ∂E1[uH(·)|·]∂WB,1

∣∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)


By the relation

∂E1[uH(WH,2) |WB,1 ]

∂WH,1

= − γ ·Rf,1 · E1[uH(WH,2) |WB,1 ]

we can re-arrange

∞∫

bdx,1

[∂2E1[uH(WH,2)|·]∂WB,1∂WH,1

· (x1 −RD,0) +∂2E1[uH(WH,2)|·]∂RD,1∂WH,1

· ∂R∗D,1

∂D0

+

+∂2E1[uH(WH,2)|·]

∂`1∂WH,1

· ∂`∗1

∂D0

]· (RD,0 −Rf,0) · 1

σ1

· ϕ(z1)dx1

≡ − γ · (RD,0 −Rf,0) ·Rf,1 ·∞∫

bdx,1

(∂E1[uH(WH,2)|·]

∂WB,1

· (x1 −RD,0) +

+∂E1[uH(WH,2)|·]

∂RD,1

· ∂R∗D,1

∂D0

+∂E1[uH(WH,2)|·]

∂`1

· ∂`∗D,1

∂D0

)· 1

σ1

· ϕ(z1) dx1

as we can change the order of derivatives with respect to WH,1 and any of the

variables WB,1, RD,1, and `1. Using the formulæ for g(D0) and g′(D0), respectively,

we can re-write the expression from above as

∞∫

bdx,1

[∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂WH,1

· (x1 −RD,0) +∂2E1[uH(WH,2) |WB,1 ]

∂RD,1∂WH,1

· ∂R∗D,1

∂D0

+

+∂2E1[uH(WH,2) |WB,1 ]

∂`1∂WH,1

· ∂`∗1

∂D0

]· (RD,0 −Rf,0) · 1

σ1

· ϕ(z1)dx1

= − γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

− γ · (RD,0 −Rf,0) ·Rf,1 · E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

− γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0)

Note that

d

dD0

[E1[uH(WH,2) |WB,1 ]

∣∣∣x1=bdx,1

]≡ ∂

∂D0

[−e−γWH,1Rf,1

]

holds because of (D.2.6), which is equal to

− γ · (RD,0 −Rf,0) ·Rf,1 · E1[uH(WH,2) |WB,1 ]x1=bdx,1.


Hence, the total derivative

d

dD0

[− E1[uH(WH,2) |WB,1 ]

∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

]

and the integral

∞∫

bdx,1

[∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂WH,1

· (x1 −RD,0) +∂2E1[uH(WH,2) |WB,1 ]

∂RD,1∂WH,1

· ∂R∗D,1

∂D0

+

+∂2E1[uH(WH,2) |WB,1 ]

∂`1∂WH,1

· ∂`∗1

∂D0

]· (RD,0 −Rf,0) · 1

σ1

· ϕ(z1)dx1

partially offset each other, that is we obtain

∞∫

bdx,1

[∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂WH,1

· (x1 −RD,0) +∂2E1[uH(WH,2) |WB,1 ]

∂RD,1∂WH,1

· ∂R∗D,1

∂D0

+

+∂2E1[uH(WH,2) |WB,1 ]

∂`1∂WH,1

· ∂`∗1

∂D0

]· (RD,0 −Rf,0) · 1

σ1

· ϕ(z1)dx1

+d

dD0

[− E1[uH(WH,2) |WB,1 ]

∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

]

= − γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

− γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0)

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ∂ϕ(bdz ,1)

∂D0

,

where

∂ϕ(bdz ,1)

∂D0

= ϕ(bdz ,1) · bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2> 0 .

As a consequence, the second derivative can be further simplified to yield


− 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) − γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

+

∞∫

bdx,1

[∂2E1[uH(·)|·]

∂W 2B,1

· (x1 −RD,0)2 +∂2E1[uH(·)|·]

∂R2D,1

·(∂R∗D,1∂D0

)2

+

+∂2E1[uH(·)|·]

∂`21

·(∂`∗1∂D0

)2]· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

[∂2E1[uH(·)|·]∂WB,1∂D1

(x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂D1

∂R∗D,1∂D0

+∂2E1[uH(·)|·]∂`1∂D1

∂`∗1∂D0

]·

·[∂Dd

1

∂D0

+∂Dd

1

∂RD,1

· ∂R∗D,1

∂D0

+∂Dd

1

∂`1

· ∂`∗1

∂D0

]· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂`1

· (x1 −RD,0) · ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂RD,1

· (x1 −RD,0) · ∂R∗D,1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂RD,1∂`1

· ∂R∗D,1

∂D0

· ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

(x1 −RD,0)2

[∂E1[uH(·)|·]∂RD,1

∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

∂2`∗1∂W 2

B,1

]1

σ1

ϕ(z1)dx1

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) · bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2

+(RD,0WB,0)2

(D0 +WB,0)3

∂E1[uH(·)|·]∂WB,1

∣∣∣∣x1=bdx,1

+

[∂E1[uH(·)|·]∂RD,1

∂R∗D,1∂WB,1

]∣∣∣∣x1=bdx,1

+

+

[∂E1[uH(·)|·]

∂`1

∂`∗1∂WB,1

]∣∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)


Because of the implicit function theorem and because the direct dependence on D0

is exclusively based on the bank’s and the household’s wealth in t = 1, we obtain

the relation

∂Ds1

∂D0

+∂Ds

1

∂RD,1

∂R∗D,1∂D0

+∂Ds

1

∂`1

∂`∗1∂D0

=

∂2E1[uH(·)|·]∂WB,1∂D1

(x1−RD,0) + ∂2E1[uH(·)|·]∂WH,1∂D1

(RD,0−Rf,0)

− ∂2E1[uH(·)|·]∂D2

1

−∂2E1[uH(·)|·]∂RD,1∂D1

∂2E1[uH(·)|·]∂D2

1

· ∂R∗D,1

∂D0

−∂2E1[uH(·)|·]∂`1∂D1

∂2E1[uH(·)|·]∂D2

1

· ∂`∗1

∂D0

.

But because of

∂2E1[uH(·)|·]∂WH,1∂D1

= − γ ·Rf,1 ·∂E1[uH(·)|·]

∂D1

= 0

given the optimal choices in t = 1, we obtain

∂Ds1

∂D0

+∂Ds

1

∂RD,1

· ∂R∗D,1

∂D0

+∂Ds

1

∂`1

· ∂`∗1

∂D0

= −∂2E1[uH(WH,2)|·]

∂WB,1∂D1(x1 −RD,0) +

∂2E1[uH(WH,2)|·]∂RD,1∂D1

∂R∗D,1∂D0

+∂2E1[uH(WH,2)|·]

∂`1∂D1

∂`∗1∂D0

∂2E1[uH(WH,2)|WB,1]

∂D21

,

resulting in

∞∫

bdx,1

[∂2E1[uH(·)|·]∂WB,1∂D1

(x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂D1

∂R∗D,1∂D0

+∂2E1[uH(·)|·]∂`1∂D1

∂`∗1∂D0

]·

·[∂Dd

1

∂D0

+∂Dd

1

∂RD,1

· ∂R∗D,1

∂D0

+∂Dd

1

∂`1

· ∂`∗1

∂D0

]· 1

σ1

· ϕ(z1)dx1

=

∞∫

bdx,1

− 1∂2E1[uH(·)|·]

∂D21

·[∂2E1[uH(·)|·]∂WB,1∂D1

· (x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂D1

· ∂R∗D,1

∂D0

+∂2E1[uH(·)|·]∂`1∂D1

· ∂`∗1

∂D0

]2

· 1

σ1

· ϕ(z1)dx1 > 0 .


Thus, the second derivative can be further simplified to

− 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) − γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

+

∞∫

bdx,1

[∂2E1[uH(·)|·]

∂W 2B,1

· (x1 −RD,0)2 +∂2E1[uH(·)|·]

∂R2D,1

·(∂R∗D,1∂D0

)2

+

+∂2E1[uH(·)|·]

∂`21

·(∂`∗1∂D0

)2]· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

− 1∂2E1[uH(·)|·]

∂D21

·[∂2E1[uH(·)|·]∂WB,1∂D1

· (x1 −RD,0) +∂2E1[uH(·)|·]∂RD,1∂D1

· ∂R∗D,1

∂D0

+∂2E1[uH(·)|·]∂`1∂D1

· ∂`∗1

∂D0

]2

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂`1

· (x1 −RD,0) · ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂WB,1∂RD,1

· (x1 −RD,0) · ∂R∗D,1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

2 · ∂2E1[uH(WH,2) |WB,1 ]

∂RD,1∂`1

· ∂R∗D,1

∂D0

· ∂`∗1

∂D0

· 1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

(x1 −RD,0)2

[∂E1[uH(·)|·]∂RD,1

∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

∂2`∗1∂W 2

B,1

]1

σ1

ϕ(z1)dx1

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) · bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2

+(RD,0WB,0)2

(D0 +WB,0)3

∂E1[uH(·)|·]∂WB,1

∣∣∣∣x1=bdx,1

+

[∂E1[uH(·)|·]∂RD,1

∂R∗D,1∂WB,1

]∣∣∣∣x1=bdx,1

+

+

[∂E1[uH(·)|·]

∂`1

∂`∗1∂WB,1

]∣∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

The remaining integrand is a quadratic form which can be illustrated by

appropriately defining a 3× 3 matrix D. To do so, let us first consider the following

2×2 matrices associated with second partial derivatives of the household’s expected

utility in t = 1. All partial derivatives are taken with respect to parameters and


variables, respectively, that are affected by the household’s choice of deposits in

t = 0. Only the associated determinants are of interest. Therefore we define:

DWB,1,WB,1:=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂W 2B,1

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂D21

∣∣∣∣∣∣

(D.2.10)

DRD,1,RD,1 :=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂R2D,1

∂2E1[uH(WH,2)|WB,1 ]

∂RD,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂RD,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂D21

∣∣∣∣∣∣

(D.2.11)

D`1,`1 :=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂`21

∂2E1[uH(WH,2)|WB,1 ]

∂`1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂`1∂D1

∂2E1[uH(WH,2)]

∂D21

∣∣∣∣∣∣

(D.2.12)

DWB,1,RD,1 :=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂RD,1∂2E1[uH(WH,2)|WB,1 ]

∂D21

∂2E1[uH(WH,2)|WB,1 ]

∂RD,1∂D1

∣∣∣∣∣∣

(D.2.13)

DWB,1,`1 :=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂WB,1∂`1∂2E1[uH(WH,2)|WB,1 ]

∂D21

∂2E1[uH(WH,2)|WB,1 ]

∂`1∂D1

∣∣∣∣∣∣

(D.2.14)

DRD,1,`1 :=

∣∣∣∣∣∣

∂2E1[uH(WH,2)|WB,1 ]

∂RD,1∂D1

∂2E1[uH(WH,2)|WB,1 ]

∂RD,1∂`1∂2E1[uH(WH,2)|WB,1 ]

∂D21

∂2E1[uH(WH,2)|WB,1 ]

∂`1∂D1

∣∣∣∣∣∣

(D.2.15)

Next, let us define the following symmetric 3 × 3 matrix D whose entries are the

determinants from above:

D =

DWB,1,WB,1DWB,1,RD,1 DWB,1,`1

DWB,1,RD,1 DRD,1,RD,1 DRD,1,`1

DWB,1,`1 DRD,1,`1 D`1,`1

(D.2.16)

Furthermore let

∆>1 =

(1,

∂R∗D,1∂WB,1

,∂`∗1

∂WB,1

)

∆>E1[uH(·)|·] =

(∂E1 [uH(·)|·]∂WB,1

,∂E1 [uH(·)|·]

∂RD,1

,∂E1 [uH(·)|·]

∂`1

).

Thus we obtain:

∂2E0

[E1

[uH(WH,2)

∣∣∣WB,1

]]

∂D20

∣∣∣∣∣∣ ∂E0[E1[uH (·)|·]]∂D0

=0

=


− 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) + h′′(D0)

− γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

+

∞∫

bdx,1

(x1 −RD,0)2

∂2E1[uH(·)|·]∂D2

1

·∆>1 ·D ·∆1 ·1

σ1

· ϕ(z1)dx1

+

∞∫

bdx,1

(x1 −RD,0)2

[∂E1[uH(·)|·]∂RD,1

∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

∂2`∗1∂W 2

B,1

]1

σ1

ϕ(z1)dx1

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) · bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2

+(RD,0WB,0)2

(D0 +WB,0)3· ∆>1 ·∆E1[uH(·)|·]

∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) .

The expressions in the first line are negative given D0 fulfills the first-order condition.

However, the expressions in the second and fifth line are always positive and it is

not clear if and how they offset each other. The integral in the third line is negative

if the matrix D is positive semi-definite. To be able to offset the impact of terms

with negative signs, it is desirable to have D positive definite.

Moreover, the definiteness of D also influences the monotonicity of the vector

product ∆T1 ·∆E1[uH(·)|·] with respect to x1. It decreases in x1 if

d[∆>1 ·∆E1[uH(·)|·]

]

dWB,1

=1

∂2E1[uH(·)|·]∂D2

1

·∆>1 ·D ·∆1 +

+

[∂E1[uH(·)|·]∂RD,1

· ∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

· ∂2`∗1

∂W 2B,1

]

is negative.

Due to (D.2.4), the expression −2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) can be rephrased


by

− 2 · γ · (RD,0 −Rf,0) ·Rf,1 ·∞∫

bdx,1

(x1 −RD,0) ·∆>1 ·∆E1[uH(·)|·] ·1

σ1

· ϕ(z1) dx1

+ 2 · γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

+ 2 · γ · (RD,0 −Rf,0) ·Rf,1 · E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

resulting in

∂2E0[uH(WH,2) |WB,1 ]

∂D20

∣∣∣∣∣∂E0[uH (·)|·]

∂D0=0

=

+ γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0) + h′′(D0)

+

∞∫

bdx,1

(x1 −RD,0)2

∂2E1[uH(·)|·]∂D2

1

·∆>1 ·D ·∆1 ·1

σ1

· ϕ(z1)dx1

− 2 · γ · (RD,0 −Rf,0) ·Rf,1 ·∞∫

bdx,1

(x1 −RD,0) ·∆>1 ·∆E1[uH(·)|·] ·1

σ1

· ϕ(z1) dx1

+

∞∫

bdx,1

(x1 −RD,0)2

[∂E1[uH(·)|·]∂RD,1

∂2R∗D,1∂W 2

B,1

+∂E1[uH(·)|·]

∂`1

∂2`∗1∂W 2

B,1

]1

σ1

ϕ(z1)dx1

−[bdz ,1RD,0WB,0

σ1(D0 +WB,0)2− 2γ(RD,0 −Rf,0)Rf,1

]·E1[uH(·)|·]|x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

Thus, the positive term −γ2 · (RD,0−Rf,0)2 ·R2f,1 · g(D0) is offset by −2 · γ · (RD,0−

Rf,0) ·Rf,1 · g′(D0), which is negative, given that the first-order condition is fulfilled.

In other words,

− 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) + h′′(D0)

− γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)


is negative.

Hence, in order to obtain a negative sign for the whole second derivative the following

conditions must hold:

0 > − 2 · γ · (RD,0 −Rf,0) ·Rf,1 · g′(D0) + h′′(D0)

− γ2 · (RD,0 −Rf,0)2 ·R2f,1 · g(D0)

− E1[uH(WH,2) |WB,1 ]∣∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1) · bdz ,1 ·RD,0WB,0

σ1(D0 +WB,0)2

+(RD,0WB,0)2

(D0 +WB,0)3· ∆>1 ·∆E1[uH(·)|·]

∣∣x1=bdx,1

· 1

σ1

· ϕ(bdz ,1)

0 >d[∆>1 ·∆E1[uH(·)|·]

]

dWB,1

.

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Lebenslauf

Seit Nov. 2009 Wirtschaftsministerium Baden-Wurttemberg

Referent

Abteilung 3/Referat 35

Aug. 2004 bis Juli 2009 Universitat Mannheim

Wissenschaftlicher Mitarbeiter

Lehrstuhl fur ABWL und Finanzierung

Okt. 1998 bis Juni 2004 Universitat Karlsruhe (TH)

Studium der technischen Volkswirtschaftslehre

Abschluss: technischer Diplom-Volkswirt

Sept. 2001 bis Juni 2002 Lunds Universitet/Universitat Lund (Schweden)

Auslandsstudium

Juni 1998 Abitur am Nikolaus-Kopernikus-Gymnasium Weißenhorn

19. Aug. 1978 Geboren in Ulm/Donau

345

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